%------------------------------------------------------------------------------
% File     : ITP262^1 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_DeleteCorrectness 00209_010241
% 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_00209_010241 [Des22]

% Status   : Theorem
% Rating   : 0.50 v9.0.0, 0.80 v8.2.0, 0.77 v8.1.0
% Syntax   : Number of formulae    : 11165 (5808 unt; 916 typ;   0 def)
%            Number of atoms       : 28393 (12566 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 116376 (2919   ~; 529   |;1869   &;100321   @)
%                                         (   0 <=>;10738  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :   95 (  94 usr)
%            Number of type conns  : 3561 (3561   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  825 ( 822 usr;  59 con; 0-8 aty)
%            Number of variables   : 25514 (1740   ^;23048   !; 726   ?;25514   :)
% 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 07:48:59.771
%------------------------------------------------------------------------------
% Could-be-implicit typings (94)
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% Explicit typings (822)
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thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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thf(sy_c_Complex_Ocomplex_OComplex,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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thf(sy_c_Complex_Ocsqrt,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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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_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__Real__Oreal,type,
    set_real2: list_real > set_real ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Onth_001t__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__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__Nat__Onat_M_Eo_J,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

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

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__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

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

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

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

thf(sy_c_List_Oproduct_001_Eo_001t__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__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__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__Real__Oreal,type,
    replicate_real: nat > real > list_real ).

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

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

thf(sy_c_List_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_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__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__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__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__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__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_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    bot_bot_set_set_int: set_set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

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,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

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,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Real__Oreal_J,type,
    ord_le4104064031414453916r_real: filter_real > filter_real > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: num > num > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Int__Oint_J,type,
    ord_max_set_int: set_int > set_int > set_int ).

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_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

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

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

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

thf(sy_c_Product__Type_OPair_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

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,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

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,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,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_001_Eo,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

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_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

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

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_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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

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

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

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

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

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

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    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__Complex__Ocomplex,type,
    insert_complex: complex > set_complex > set_complex ).

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__Num__Onum,type,
    insert_num: num > set_num > set_num ).

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

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

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__VEBT____Definitions__OVEBT,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

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_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__Int__Oint_J,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

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_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__Extended____Nat__Oenat,type,
    set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).

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

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

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

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

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

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

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

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

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__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__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_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__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__Real__Oreal,type,
    tan_real: real > real ).

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

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

thf(sy_c_VEBT__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__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_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__Num__Onum_Mt__Num__Onum_J,type,
    accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__List__Olist_It__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__Int__Oint_Mt__Int__Oint_J,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

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

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

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_maxs____,type,
    maxs: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary____,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10208)
thf(fact_0__C1_C,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "1"
thf(fact_1__092_060open_062both__member__options_Asummary_Amaxs_092_060close_062,axiom,
    vEBT_V8194947554948674370ptions @ summary @ maxs ).

% \<open>both_member_options summary maxs\<close>
thf(fact_2__C8_C,axiom,
    na = m ).

% "8"
thf(fact_3_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_4_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_5__092_060open_062invar__vebt_A_ItreeList_091high_Ax_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_Ax_An_J_A_Ilow_Ax_An_J_093_A_B_Amaxs_J_An_092_060close_062,axiom,
    vEBT_invar_vebt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ maxs ) @ na ).

% \<open>invar_vebt (treeList[high x n := vebt_delete (treeList ! high x n) (low x n)] ! maxs) n\<close>
thf(fact_6_xnotmi,axiom,
    xa != mi ).

% xnotmi
thf(fact_7_hlist,axiom,
    ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( vEBT_VEBT_high @ xa @ na ) )
    = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ).

% hlist
thf(fact_8_nnvalid,axiom,
    vEBT_invar_vebt @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) @ na ).

% nnvalid
thf(fact_9_ninNullc,axiom,
    vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ).

% ninNullc
thf(fact_10_bb,axiom,
    ( ( maxs
     != ( vEBT_VEBT_high @ xa @ na ) )
    & ( ord_less_nat @ maxs @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ) ).

% bb
thf(fact_11_True,axiom,
    xa = ma ).

% True
thf(fact_12_nothlist,axiom,
    ! [I: nat] :
      ( ( I
       != ( vEBT_VEBT_high @ xa @ na ) )
     => ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ I )
          = ( nth_VEBT_VEBT @ treeList @ I ) ) ) ) ).

% nothlist
thf(fact_13__C4_C,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X2 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).

% "4"
thf(fact_14_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ N2 ) ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) ) ) ).

% in_children_def
thf(fact_15_newsummvalid,axiom,
    vEBT_invar_vebt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) @ m ).

% newsummvalid
thf(fact_16__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_17_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_18_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_19_list__update__id,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_20_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_21_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
        = ( nth_int @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_22_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
        = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_23__092_060open_062treeList_A_B_Ahigh_Ax_An_A_092_060in_062_Aset_AtreeList_092_060close_062,axiom,
    member_VEBT_VEBT @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( set_VEBT_VEBT2 @ treeList ) ).

% \<open>treeList ! high x n \<in> set treeList\<close>
thf(fact_24_allvalidinlist,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) ) )
     => ( vEBT_invar_vebt @ X4 @ na ) ) ).

% allvalidinlist
thf(fact_25__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_26__C111_C,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ I2 ) @ X2 ) )
        = ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) @ I2 ) ) ) ).

% "111"
thf(fact_27__C4_OIH_C_I2_J,axiom,
    ! [X: nat] : ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ summary @ X ) @ m ) ).

% "4.IH"(2)
thf(fact_28_list__update__overwrite,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I @ Y )
      = ( list_u1324408373059187874T_VEBT @ Xs @ I @ Y ) ) ).

% list_update_overwrite
thf(fact_29_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_30__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_31__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_32__C0_C,axiom,
    ! [X4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X4 @ na ) ) ).

% "0"
thf(fact_33_hlbound,axiom,
    ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).

% hlbound
thf(fact_34__C4_Ohyps_C_I7_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "4.hyps"(7)
thf(fact_35__C4_Ohyps_C_I8_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

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

% "2"
thf(fact_37__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 ) ) )
          & ! [Y2: nat] :
              ( ( ( ( vEBT_VEBT_high @ Y2 @ na )
                  = I2 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ Y2 @ na ) ) )
             => ( ( ord_less_nat @ mi @ Y2 )
                & ( ord_less_eq_nat @ Y2 @ ma ) ) ) ) ) ) ).

% "7"
thf(fact_38_list__update__swap,axiom,
    ! [I: nat,I3: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X5: vEBT_VEBT] :
      ( ( I != I3 )
     => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I3 @ X5 )
        = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I3 @ X5 ) @ I @ X ) ) ) ).

% list_update_swap
thf(fact_39_yhelper,axiom,
    ! [Y: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ Y @ na ) ) @ ( vEBT_VEBT_low @ Y @ na ) )
     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Y @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ord_less_nat @ mi @ Y )
          & ( ord_less_eq_nat @ Y @ ma )
          & ( ord_less_nat @ ( vEBT_VEBT_low @ Y @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ) ) ).

% yhelper
thf(fact_40__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 ) ) )
        & ! [Y2: nat] :
            ( ( ( ( vEBT_VEBT_high @ Y2 @ na )
                = I2 )
              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ Y2 @ na ) ) )
           => ( ( ord_less_nat @ mi @ Y2 )
              & ( ord_less_eq_nat @ Y2 @ ma ) ) ) ) ) ).

% "7b"
thf(fact_41_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_42_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_43_newlistlength,axiom,
    ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% newlistlength
thf(fact_44__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_45_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_46_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_47_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_48_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
    ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X3: real] : ( member_real @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A2: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X3: complex] : ( member_complex @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X6: product_prod_int_int] :
          ( ( P @ X6 )
          = ( Q @ X6 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_62_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X6: complex] :
          ( ( P @ X6 )
          = ( Q @ X6 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_63_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X6: set_nat] :
          ( ( P @ X6 )
          = ( Q @ X6 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_64_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X6: nat] :
          ( ( P @ X6 )
          = ( Q @ X6 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_65_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ( P @ X6 )
          = ( Q @ X6 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_66_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_67_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_68_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_69_numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% numeral_less_iff
thf(fact_70_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_71_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_72__C3_C,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "3"
thf(fact_73_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_74_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_75_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs2 )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs2 )
             => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).

% min_in_set_def
thf(fact_76_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X3: nat] :
          ( ( member_nat @ X3 @ Xs2 )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs2 )
             => ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).

% max_in_set_def
thf(fact_77_inthall,axiom,
    ! [Xs: list_complex,P: complex > $o,N: nat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ ( set_complex2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
       => ( P @ ( nth_complex @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_78_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N: nat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ ( set_real2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_79_inthall,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
      ( ! [X6: set_nat] :
          ( ( member_set_nat @ X6 @ ( set_set_nat2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_80_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_81_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N: nat] :
      ( ! [X6: $o] :
          ( ( member_o @ X6 @ ( set_o2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_82_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N: nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_83_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N: nat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
         => ( P @ X6 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% inthall
thf(fact_84_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_85_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_86_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_87_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_88_numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N ) )
      = ( M = N ) ) ).

% numeral_eq_iff
thf(fact_89_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(87)
thf(fact_90__C12_C,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% "12"
thf(fact_91_inrg,axiom,
    ( ( ord_less_eq_nat @ mi @ xa )
    & ( ord_less_eq_nat @ xa @ ma ) ) ).

% inrg
thf(fact_92_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_93_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_94_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_95_numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% numeral_le_iff
thf(fact_96_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_97_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_98_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_99_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_100_numeral__plus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_101_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_102_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_103_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_104_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_105_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_106_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_107_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_108_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_list_update
thf(fact_109_length__list__update,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_list_update
thf(fact_110_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_111_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_112_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(78)
thf(fact_113_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_114__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_115_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_116_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
     => ( ( list_update_o @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_117_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_118_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X )
        = Xs ) ) ).

% list_update_beyond
thf(fact_119_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_120_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_121_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_122_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_123_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_124_set__swap,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_125_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
          = ( set_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_126_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_127_set__swap,axiom,
    ! [I: nat,Xs: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
          = ( set_int2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_128_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_129_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_130_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_131_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_132_neq__if__length__neq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
       != ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_133_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
       != ( size_size_list_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_134_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_135_neq__if__length__neq,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
       != ( size_size_list_int @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_136_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_137_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_138_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_139_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: 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 @ Xs ) ) ).

% length_induct
thf(fact_140_length__induct,axiom,
    ! [P: list_o > $o,Xs: 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 @ Xs ) ) ).

% length_induct
thf(fact_141_length__induct,axiom,
    ! [P: list_nat > $o,Xs: 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 @ Xs ) ) ).

% length_induct
thf(fact_142_length__induct,axiom,
    ! [P: list_int > $o,Xs: 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 @ Xs ) ) ).

% length_induct
thf(fact_143_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_144_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_Bit0
thf(fact_145_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_Bit0
thf(fact_146_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_Bit0
thf(fact_147_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_Bit0
thf(fact_148_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I4 )
              = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_149_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I4 )
              = ( nth_o @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_150_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_151_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I4 )
              = ( nth_int @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_152_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X2: vEBT_VEBT] : ( P @ I5 @ X2 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P @ I5 @ ( nth_VEBT_VEBT @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_153_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X2: $o] : ( P @ I5 @ X2 ) ) )
      = ( ? [Xs2: list_o] :
            ( ( ( size_size_list_o @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P @ I5 @ ( nth_o @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_154_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X2: nat] : ( P @ I5 @ X2 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P @ I5 @ ( nth_nat @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_155_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K )
           => ? [X2: int] : ( P @ I5 @ X2 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K )
               => ( P @ I5 @ ( nth_int @ Xs2 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_156_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : ( Y4 = Z2 ) )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I5 )
                = ( nth_VEBT_VEBT @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_157_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_o,Z2: list_o] : ( Y4 = Z2 ) )
    = ( ^ [Xs2: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs2 ) )
             => ( ( nth_o @ Xs2 @ I5 )
                = ( nth_o @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_158_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I5 )
                = ( nth_nat @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_159_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_int,Z2: list_int] : ( Y4 = Z2 ) )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I5 )
                = ( nth_int @ Ys3 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_160_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_161_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_162_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_163_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_164_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs ) )
           => ( P @ ( nth_o @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_165_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_166_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_167_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: complex > $o,X: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) )
         => ( P @ ( nth_complex @ Xs @ I4 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_168_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I4 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_169_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P @ ( nth_set_nat @ Xs @ I4 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_170_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_171_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X: $o] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I4 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_172_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I4 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_173_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I4 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_174_in__set__conv__nth,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s3451745648224563538omplex @ Xs ) )
            & ( ( nth_complex @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_175_in__set__conv__nth,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_176_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_177_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_178_in__set__conv__nth,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_179_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_180_in__set__conv__nth,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I5 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_181_list__ball__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X6: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X6 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_182_list__ball__nth,axiom,
    ! [N: nat,Xs: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( ! [X6: $o] :
            ( ( member_o @ X6 @ ( set_o2 @ Xs ) )
           => ( P @ X6 ) )
       => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_183_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
           => ( P @ X6 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_184_list__ball__nth,axiom,
    ! [N: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
           => ( P @ X6 ) )
       => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_185_nth__mem,axiom,
    ! [N: nat,Xs: list_complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
     => ( member_complex @ ( nth_complex @ Xs @ N ) @ ( set_complex2 @ Xs ) ) ) ).

% nth_mem
thf(fact_186_nth__mem,axiom,
    ! [N: nat,Xs: list_real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).

% nth_mem
thf(fact_187_nth__mem,axiom,
    ! [N: nat,Xs: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_188_nth__mem,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% nth_mem
thf(fact_189_nth__mem,axiom,
    ! [N: nat,Xs: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_190_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_191_nth__mem,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).

% nth_mem
thf(fact_192_set__update__memI,axiom,
    ! [N: nat,Xs: list_complex,X: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_193_set__update__memI,axiom,
    ! [N: nat,Xs: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_194_set__update__memI,axiom,
    ! [N: nat,Xs: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_195_set__update__memI,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_196_set__update__memI,axiom,
    ! [N: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_197_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_198_set__update__memI,axiom,
    ! [N: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_199_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_200_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X ) @ J )
        = ( ( ( I = J )
           => X )
          & ( ( I != J )
           => ( nth_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_201_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_202_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_203_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_204_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ( list_update_o @ Xs @ I @ X )
          = Xs )
        = ( ( nth_o @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_205_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_206_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_207_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X6 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_208_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_209_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_210_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_211__C9_C,axiom,
    ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = na ) ).

% "9"
thf(fact_212_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_213_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_214_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231__C11_C,axiom,
    ord_less_eq_nat @ one_one_nat @ na ).

% "11"
thf(fact_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X3: nat,N2: nat] : ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% high_def
thf(fact_242_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_243_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_244_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_245_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_246_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_247_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_248_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_249_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_250_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_251_numeral__times__numeral,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% numeral_times_numeral
thf(fact_252_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_253_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_254_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_255_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_256_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_257_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_258_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_259_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_260_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_261_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_262_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_263_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_264_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_265_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_266_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_267_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_268_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_269_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_270_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_271_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N ) )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_272_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_273_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(6)
thf(fact_274_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(71)
thf(fact_275_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_287_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_288_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_289_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_290_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_291_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_292_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_293_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_294_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_295_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_296_power__inject__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_297_power__inject__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_298_power__inject__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_299_power__inject__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N ) )
        = ( M = N ) ) ) ).

% power_inject_exp
thf(fact_300_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_301_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_302_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_303_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_304_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_305_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_306_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_307_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_308_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_309_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_310_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_311_power__strict__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_312_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_313_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_314_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_315_power__add__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_316_power__add__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_317_power__add__numeral,axiom,
    ! [A: rat,M: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_318_power__add__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_319_power__add__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% power_add_numeral
thf(fact_320_power__add__numeral2,axiom,
    ! [A: complex,M: num,N: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_321_power__add__numeral2,axiom,
    ! [A: real,M: num,N: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_322_power__add__numeral2,axiom,
    ! [A: rat,M: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_323_power__add__numeral2,axiom,
    ! [A: nat,M: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_324_power__add__numeral2,axiom,
    ! [A: int,M: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_325_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_326_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_327_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_328_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_329_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_330_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_331_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_332_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_333_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_334_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_335_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_336_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_337_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_338_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_339_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_340_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_341_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_342_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_343_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_344_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_345_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_346_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_347_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_348_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_349_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_350_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_351_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_352_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% enat_less_induct
thf(fact_353_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_354_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_364_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_365_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_366_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_367_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B2: rat] : ( times_times_rat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_368_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_369_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_370_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_371_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_372_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_373_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_374_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_375_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_376_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_377_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_378_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_379_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_380_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_381_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_382_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
      = ( X = one_one_rat ) ) ).

% one_reorient
thf(fact_383_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_384_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_385_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_386_power__one__over,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).

% power_one_over
thf(fact_387_power__one__over,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% power_one_over
thf(fact_388_power__one__over,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% power_one_over
thf(fact_389_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_390_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_391_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_392_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_393_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_394_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_395_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_396_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_397_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_398_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_399_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% add_mult_distrib
thf(fact_400_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% add_mult_distrib2
thf(fact_401_power__gt1__lemma,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_402_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_403_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_404_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_405_power__less__power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_406_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_407_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_408_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_409_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_divide
thf(fact_410_power__divide,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_divide
thf(fact_411_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_divide
thf(fact_412_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_413_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_414_power__commutes,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_commutes
thf(fact_415_power__commutes,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_commutes
thf(fact_416_power__commutes,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_commutes
thf(fact_417_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_418_power__commutes,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_commutes
thf(fact_419_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_420_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_421_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_422_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_423_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_424_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_425_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_426_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_427_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_428_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_429_power__mult,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_430_power__mult,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_431_power__mult,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_432_power__mult,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).

% power_mult
thf(fact_433_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_434_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_435_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_436_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_437_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_438_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_439_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_440_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_441_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_442_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
      = ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( set_complex2 @ Xs ) )
           => ( member_complex @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_443_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
      = ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
           => ( member_real @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_444_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B3 )
      = ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_445_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
      = ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_446_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_447_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
      = ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
           => ( member_int @ X3 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_448_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_449_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_450_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_451_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_452_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_453_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_454_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_455_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_456_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_457_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_458_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_459_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_460_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_461_power__add,axiom,
    ! [A: complex,M: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_462_power__add,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_463_power__add,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_464_power__add,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_465_power__add,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_466_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_467_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_468_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_469_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_470_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_471_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_472_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_473_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_474_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_475_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_476_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_477_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_478_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_479_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_480_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_481_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_482_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_483_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_484_one__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).

% one_le_power
thf(fact_485_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).

% one_le_power
thf(fact_486_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_487_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_488_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_489_set__update__subsetI,axiom,
    ! [Xs: list_complex,A2: set_complex,X: complex,I: nat] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_490_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_491_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
     => ( ( member_set_nat @ X @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_492_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_493_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBT @ X @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_494_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_495_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_496_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_497_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_498_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_499_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_500_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_501_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_502_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_503_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_504_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_505_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_506_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_507_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_508_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_509_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_510_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_511_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_512_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_513_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_514_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_515_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_516_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_517_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_518_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_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_527_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_528_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_529_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_530_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_531_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_532_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_power__increasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_546_power__increasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_547_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_548_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_549_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_550_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_551_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_552_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_553_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_554_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_555_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_556_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_557_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_558_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_559_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_560_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_561_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_562_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 ) )
       => ? [X6: nat] :
            ( ( P @ X6 )
            & ! [Y2: nat] :
                ( ( P @ Y2 )
               => ( ord_less_eq_nat @ Y2 @ X6 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_563_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_564_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_565_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_566_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_567_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_568_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_569_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_570_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_571_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_572_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_573_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_584_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_585_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_586_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_587_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_588_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_589_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_590_power2__eq__square,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A @ A ) ) ).

% power2_eq_square
thf(fact_591_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_592_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_593_power__even__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_594_power__even__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_595_power__even__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_596_power__even__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_597_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_598_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_599_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_600_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_601_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_602_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_603_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_604_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_605_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_606_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_607_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_608_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_609_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_610_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_611_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_612_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_613_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_614_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_615_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_616_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_617_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_618_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_619_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_620_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_621_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_622_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_623_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_624_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_625_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_626_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_627_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_628_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_629_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_630_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_631_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_632_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_633_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_634_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_635_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_636_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_637_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_638_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_639_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_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

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

% add_strict_right_mono
thf(fact_670_add__less__imp__less__left,axiom,
    ! [C: 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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M3 @ N2 )
          & ( M3 != N2 ) ) ) ) ).

% nat_less_le
thf(fact_679_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_680_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
          ( ( ord_less_nat @ M3 @ N2 )
          | ( M3 = N2 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_681_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_682_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_683_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_684_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_685_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_686_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_687_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_688_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_689_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_690_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_691_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% less_add_eq_less
thf(fact_692_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_693_le__add1,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).

% le_add1
thf(fact_694_le__add2,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).

% le_add2
thf(fact_695_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% add_leD1
thf(fact_696_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_697_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N3: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_698_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_699_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_700_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_701_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_702_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_703_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 )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_704_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 )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N3: nat] :
          ( ( ord_less_nat @ M4 @ N3 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_722_set__vebt__set__vebt_H__valid,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ T )
        = ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_723_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_724_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_725_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_726_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_727_div__exp__eq,axiom,
    ! [A: nat,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_728_div__exp__eq,axiom,
    ! [A: int,M: nat,N: 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 ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% div_exp_eq
thf(fact_729_field__less__half__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_730_field__less__half__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_731_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_732_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_733_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_734_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_735_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_736_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_737_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_751_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_752_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_753_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_754_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_755_power__mult__numeral,axiom,
    ! [A: nat,M: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_756_power__mult__numeral,axiom,
    ! [A: real,M: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_757_power__mult__numeral,axiom,
    ! [A: int,M: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_758_power__mult__numeral,axiom,
    ! [A: complex,M: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_759_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ X @ Z3 ) )
           => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).

% succ_member
thf(fact_760_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z3 )
              & ( ord_less_nat @ Z3 @ X ) )
           => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).

% pred_member
thf(fact_761_two__realpow__ge__one,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% two_realpow_ge_one
thf(fact_762_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_763_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_764_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_765_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_766_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_767_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_768_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_769_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_770_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_771_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X4 ) ).

% linordered_field_no_lb
thf(fact_772_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X4 ) ).

% linordered_field_no_lb
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_799_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_800_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_801_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_802_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_803_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_804_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_805_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_806_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_807_div__le__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).

% div_le_dividend
thf(fact_808_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).

% div_le_mono
thf(fact_809_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_810_less__1__mult,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_811_less__1__mult,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_812_less__1__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_813_less__1__mult,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).

% less_1_mult
thf(fact_814_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_815_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_816_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_817_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_818_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_819_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_820_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_821_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_822_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_823_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_824_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_825_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_826_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_827_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_828_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_829_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_830_numeral__Bit0__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit0_div_2
thf(fact_831_field__sum__of__halves,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_832_field__sum__of__halves,axiom,
    ! [X: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X ) ).

% field_sum_of_halves
thf(fact_833_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_834_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_835_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_836_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X3: nat,N2: nat] : ( modulo_modulo_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% low_def
thf(fact_837__092_060open_062Some_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_Ax_An_J_J_092_060close_062,axiom,
    ( ( some_nat @ maxs )
    = ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).

% \<open>Some maxs = vebt_maxt (vebt_delete summary (high x n))\<close>
thf(fact_838_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_839_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_840_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_841_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_842_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_843_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_real @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_844_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_rat @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_845_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_846_power__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( numeral_numeral_int @ ( pow @ K @ L2 ) ) ) ).

% power_numeral
thf(fact_847_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_848_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_849_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_850_False,axiom,
    ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) )
   != none_nat ) ).

% False
thf(fact_851_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_852_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_853_deg__not__0,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% deg_not_0
thf(fact_854_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_855_power__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_856_maxt__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_857_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_858_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_859_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_860_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_861_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_862_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_863_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_864_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_865_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_866_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_867_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_868_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_869_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_870_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_871_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_872_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_873_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N @ K ) )
      = ( ( M = N )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_874_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_875_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_876_i0__less,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
      = ( N != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_877_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_878_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_879_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_880_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_881_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_882_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_883_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_884_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_885_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_886_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_887_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_888_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_889_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_890_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_891_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_892_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_893_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_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_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_905_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_906_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_907_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_908_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_909_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_910_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_911_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_912_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_913_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_914_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_915_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_937_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_938_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_939_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_940_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_941_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_942_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_943_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_944_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_945_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_946_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_947_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_948_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_949_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_950_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_951_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_952_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_953_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_954_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_955_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_956_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_957_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_958_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_959_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_960_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_961_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_962_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_963_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_964_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_965_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_966_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_967_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_968_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_969_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_970_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_971_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_972_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_973_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_974_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_975_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_976_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_977_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_978_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_979_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_980_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_981_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_982_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_983_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_984_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_985_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_986_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_987_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_988_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_989_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_990_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_991_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_992_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_993_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_994_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_995_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_1013_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_1014_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_1015_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_1016_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_1017_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_1018_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_1019_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_1020_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_1021_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_1022_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_1023_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_1024_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_1025_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_1026_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_1027_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_1028_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_1029_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_1030_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_1031_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_1032_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_1033_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1034_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_1035_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% mult_less_cancel2
thf(fact_1036_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1037_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1038_div__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_1039_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1040_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_1041_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1042_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1043_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_1044_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_1045_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
          = ( divide_divide_nat @ M @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_1046_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = M ) ) ).

% mod_less
thf(fact_1047_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( 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 ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_1048_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_1049_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_1050_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_1051_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_1052__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_Ax_An_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxs: nat] :
        ( ( some_nat @ Maxs )
       != ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).

% \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt (vebt_delete summary (high x n)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1053_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( 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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1105_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1106_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1164_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1165_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1166_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1167_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_1168_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_1169_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_1170_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_1171_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_1172_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_1173_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_1174_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_1175_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_1176_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_1177_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_1178_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_1179_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_1180_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_1201_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_1202_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_1203_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_1204_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_1205_power__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( power_power_rat @ A @ N )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1206_power__eq__0__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( ( power_power_nat @ A @ N )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1207_power__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( power_power_real @ A @ N )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1208_power__eq__0__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ( power_power_int @ A @ N )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1209_power__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( power_power_complex @ A @ N )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% power_eq_0_iff
thf(fact_1210_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_1211_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_1212_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_1213_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_1214_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_1215_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_1216_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_1217_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_1218_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_1219_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_1220_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_1221_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_1222_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_1223_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_1224_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_1225_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_1226_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_1227_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_1228_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_1229_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_1230_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_1231_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_1232_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_1233_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_1234_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_1235_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_1236_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_1237_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_1238_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_1239_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_1240_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% mult_le_cancel2
thf(fact_1241_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1242_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_1243_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_1244_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_1245_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_1246_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_1247_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_1248_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_1249_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_1250_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_1251_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_1252__C10_C,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).

% "10"
thf(fact_1253_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_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_1274_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_1275_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_1276_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_1277_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_1278_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_1279_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_1280_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_1281_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_1282_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_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1291_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1292_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1293_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_1294_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).

% lesseq_shift
thf(fact_1295_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_1296_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_1297_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_1298_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_1299_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_1300_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_1301_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_1302_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_1303_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1304_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1305_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1306_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N: 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 @ N ) )
          = ( ord_less_nat @ N @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_1307_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_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_1315_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_1316_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_1317_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_1318_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_1319_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_1320_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_1321_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_1322_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_1323_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_1324_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_1325_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_1326_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1327_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1328_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1329_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_1330_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1331_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1332_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1333_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N: 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 @ N ) )
          = ( ord_less_eq_nat @ N @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_1334_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_1335_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_1336_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_1337_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1338_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1339_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1340_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_1341_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N3 ) ) ) ).

% real_arch_pow
thf(fact_1342_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y3: real] :
          ( ( ord_less_real @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% less_eq_real_def
thf(fact_1343_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ one_one_real )
       => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X @ N3 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_1344_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X4: real] : ( member_real @ X4 @ S2 )
     => ( ? [Z4: real] :
          ! [X6: real] :
            ( ( member_real @ X6 @ S2 )
           => ( ord_less_eq_real @ X6 @ Z4 ) )
       => ? [Y5: real] :
            ( ! [X4: real] :
                ( ( member_real @ X4 @ S2 )
               => ( ord_less_eq_real @ X4 @ Y5 ) )
            & ! [Z4: real] :
                ( ! [X6: real] :
                    ( ( member_real @ X6 @ S2 )
                   => ( ord_less_eq_real @ X6 @ Z4 ) )
               => ( ord_less_eq_real @ Y5 @ Z4 ) ) ) ) ) ).

% complete_real
thf(fact_1345_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1346_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_1347_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_1348_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_1349_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_1350_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_1351_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_1352_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_1353_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1354_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_1355_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_1356_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_1357_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1358_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_1359_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1360_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1361_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1362_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1363_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_1364_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1365_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1366_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1367_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1368_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: complex,Z2: complex] : ( Y4 = Z2 ) )
    = ( ^ [A3: complex,B2: complex] :
          ( ( minus_minus_complex @ A3 @ B2 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1369_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: real,Z2: real] : ( Y4 = Z2 ) )
    = ( ^ [A3: real,B2: real] :
          ( ( minus_minus_real @ A3 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1370_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
    = ( ^ [A3: rat,B2: rat] :
          ( ( minus_minus_rat @ A3 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1371_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( minus_minus_int @ A3 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1372_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1373_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1374_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1375_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1376_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_1377_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_1378_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_1379_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1380_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).

% mod_if
thf(fact_1381_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( modulo_modulo_nat @ M @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_1382_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_1383_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_1384_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_1385_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1386_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1387_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1388_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1389_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1390_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1391_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_1392_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_1393_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1394_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X6: real] :
            ( ( ord_less_real @ zero_zero_real @ X6 )
            & ( ( power_power_real @ X6 @ N )
              = A )
            & ! [Y2: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y2 )
                  & ( ( power_power_real @ Y2 @ N )
                    = A ) )
               => ( Y2 = X6 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_1395_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ( ( power_power_real @ R2 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_1396_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1397_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_1398_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_1399_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1400_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_1401_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_1402_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1403_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1404_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1405_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_1406_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_1407_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_1408_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_1409_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_1410_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_1411_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_1412_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_1413_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_1414_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_1415_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_1416_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_1417_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_1418_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_1419_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_1420_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_1421_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_1422_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_1423_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_1424_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_1425_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_1426_diff__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_1427_diff__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_1428_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_1429_divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_1430_divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_1431_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_1432_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).

% modulo_nat_def
thf(fact_1433_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_1434_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_1435_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_1436_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_1437_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_1438_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_1439_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_1440_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_1441_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_rat @ zero_zero_rat @ N )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_1442_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_nat @ zero_zero_nat @ N )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_1443_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ zero_zero_real @ N )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_1444_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_int @ zero_zero_int @ N )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_1445_zero__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ zero_zero_complex @ N )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_1446_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_1447_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_1448_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_1449_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1450_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1451_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_1452_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_1453_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_1454_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_1455_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_1456_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_1457_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_1458_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_1459_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_1460_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_1461_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_1462_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_1463_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_1464_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_1465_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_1466_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_1467_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1468_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1469_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_1470_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_1471_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_1472_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_1473_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_1474_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_1475_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_1476_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1477_power__mod,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1478_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_1479_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_1480_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_1481_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_1482_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_1483_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_1484_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_1485_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_1486_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_1487_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_1488_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_1489_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_1490_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_1491_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_1492_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_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_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_1501_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_1502_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_1503_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_1504_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_1505_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_1506_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_1507_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_1508_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_1509_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_1510_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1511_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_1512_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_1513_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_1514_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_1515_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_1516_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_1517_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_1518_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_1519_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_1520_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_1521_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_1522_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_1523_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_1524_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_1525_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_1526_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_1527_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_1528_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_1529_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_1530_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_1531_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_1532_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_1533_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_1534_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_1535_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_1536_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_1537_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_1538_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_1539_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_1540_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_1541_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_1542_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_1543_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_1544_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_1545_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_1546_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_1547_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_1548_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_1549_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).

% mod_less_eq_dividend
thf(fact_1550_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_1551_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_1552_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_1553_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1554_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1555_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N @ K ) )
          = ( M = N ) ) ) ) ).

% eq_diff_iff
thf(fact_1556_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% le_diff_iff
thf(fact_1557_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1558_diff__le__mono,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).

% diff_le_mono
thf(fact_1559_diff__le__self,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).

% diff_le_self
thf(fact_1560_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_1561_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_1562_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% Nat.diff_cancel
thf(fact_1563_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_cancel2
thf(fact_1564_diff__add__inverse,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
      = M ) ).

% diff_add_inverse
thf(fact_1565_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
      = M ) ).

% diff_add_inverse2
thf(fact_1566_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).

% diff_mult_distrib
thf(fact_1567_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_1568_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_1569_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1570_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1571_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1572_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1573_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_1574_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1575_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1576_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_1577_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1578_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1579_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1580_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1581_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_1582_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_1583_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_1584_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_1585_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_1586_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_1587_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_1588_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1589_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1590_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_1591_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1592_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1593_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_1594_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_1595_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_1596_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_1597_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_1598_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_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_1625_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1626_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_1627_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1628_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1629_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1630_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1631_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1632_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1633_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1634_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1635_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1636_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1637_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1638_power__not__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_1639_power__not__zero,axiom,
    ! [A: nat,N: nat] :
      ( ( A != zero_zero_nat )
     => ( ( power_power_nat @ A @ N )
       != zero_zero_nat ) ) ).

% power_not_zero
thf(fact_1640_power__not__zero,axiom,
    ! [A: real,N: nat] :
      ( ( A != zero_zero_real )
     => ( ( power_power_real @ A @ N )
       != zero_zero_real ) ) ).

% power_not_zero
thf(fact_1641_power__not__zero,axiom,
    ! [A: int,N: nat] :
      ( ( A != zero_zero_int )
     => ( ( power_power_int @ A @ N )
       != zero_zero_int ) ) ).

% power_not_zero
thf(fact_1642_power__not__zero,axiom,
    ! [A: complex,N: nat] :
      ( ( A != zero_zero_complex )
     => ( ( power_power_complex @ A @ N )
       != zero_zero_complex ) ) ).

% power_not_zero
thf(fact_1643_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1644_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1645_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1646_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1647_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1648_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1649_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1650_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_1651_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1652_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_1653_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_1654_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1655_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1656_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1657_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1658_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1659_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_1660_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_1661_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_1662_frac__le__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_1663_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_1664_frac__less__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_1665_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_1666_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1667_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_1668_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_1669_not__iless0,axiom,
    ! [N: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_1670_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_1671_power__diff,axiom,
    ! [A: complex,N: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_1672_power__diff,axiom,
    ! [A: real,N: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_1673_power__diff,axiom,
    ! [A: rat,N: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_1674_power__diff,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_1675_power__diff,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_1676_split__mod,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I5 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_1677_i0__lb,axiom,
    ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).

% i0_lb
thf(fact_1678_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
      = ( N = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_1679_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_1680_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_1681_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_1682_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1683_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1684_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1685_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_1686_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1687_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1688_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1689_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_1690_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% mult_eq_if
thf(fact_1691_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_1692_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_1693_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_1694_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_1695_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_1696_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_1697_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1698_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1699_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_1700_scaling__mono,axiom,
    ! [U: real,V: real,R: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R )
       => ( ( ord_less_eq_real @ R @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_1701_scaling__mono,axiom,
    ! [U: rat,V: rat,R: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
       => ( ( ord_less_eq_rat @ R @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_1702_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_1703_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_1704_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_1705_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_1706_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_1707_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1708_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1709_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1710_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1711_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P2: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P2 @ ( power_power_complex @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1712_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P2: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P2 @ ( power_power_real @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1713_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P2: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P2 @ ( power_power_rat @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1714_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P2: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P2 @ ( power_power_nat @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1715_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P2: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P2 @ ( power_power_int @ P2 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_1716_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_1717_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_1718_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_1719_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_1720_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_1721_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_1722_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_1723_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_1724_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_1725_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_1726_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_1727_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_1728_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_1729_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_1730_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_1731_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_1732_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1733_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1734_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1735_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1736_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1737_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1738_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1739_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1740_power__minus__mult,axiom,
    ! [N: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_complex @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1741_power__minus__mult,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_real @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1742_power__minus__mult,axiom,
    ! [N: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1743_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1744_power__minus__mult,axiom,
    ! [N: nat,A: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_int @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_1745_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_1746_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_1747_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_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1756_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1757_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_1758_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1759_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1760_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_1761_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_1762_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_1763_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_1764_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_1765_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_1766_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_1767_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_1768_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_1769_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_1770_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_1771_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_1772_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ~ ! [S3: nat] :
              ( N
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_1773_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_1774_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_1775_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_1776_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_1777_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_1778_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_1779_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1780_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_1781_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_1782_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ ( ord_less_nat @ M @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1783_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_1784_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_1785_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_1786_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_1787_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_1788_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_1789_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X3: real] : ( plus_plus_real @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_1790_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_1791_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X3: int] : ( plus_plus_int @ X3 @ X3 ) ) ) ).

% dbl_def
thf(fact_1792_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1793_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_1794_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1795_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1796_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_1797_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_1798_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_1799_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_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_1834_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_1835_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_1836_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_1837_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_1838_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_1839_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_1840_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_1841_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_1842_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_1843_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_1844_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_1845_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_1846_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_1847_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_1848_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_1849_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_1850_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_1851_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_1852_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_1853_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_1854_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_1855_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_1856_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_1857_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_1858_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_1859_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_1860_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_1861_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_1862_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_1863_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_1864_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_1865_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1866_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_1867_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1868_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1869_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_1870_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_1871_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_1872_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_1873_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_1874_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_1875_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_1876_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_1877_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_1878_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_1879_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_1880_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_1881_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_1882_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_1883_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_1884_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_1885_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_1886_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_1887_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_1888_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_1889_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_1890_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_1891_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_1892_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_1893_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1894_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1895_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1896_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1897_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1898_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1899_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1900_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1901_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1902_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1903_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1904_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1905_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_1906_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_1907_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_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_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_1915_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_1916_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_1917_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1918_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1919_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_1920_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_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_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_1929_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_1930_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_1931_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_1932_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_1933_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_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_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_1947_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_1948_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_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_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_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1993_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1994_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1995_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1996_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1997_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1998_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1999_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2000_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2001_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2002_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2003_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2004_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2005_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2006_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2007_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_2008_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_2009_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2010_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2011_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2012_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2013_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2014_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2015_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2016_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2017_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_2018_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_2019_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_2020_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_2021_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_2022_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_2023_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2024_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_2025_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2026_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2027_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2028_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2029_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2030_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_2031_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_2032_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_2033_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_2034_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_2035_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_2036_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_2037_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_2038_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_2039_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_2040_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_2041_zero__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2042_zero__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2043_zero__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2044_zero__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_power
thf(fact_2045_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2046_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2047_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2048_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_2049_zero__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2050_zero__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2051_zero__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2052_zero__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_less_power
thf(fact_2053_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2054_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2055_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2056_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_2057_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_2058_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_2059_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_2060_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_2061_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_2062_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_2063_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_2064_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_2065_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_2066_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_2067_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_2068_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_2069_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_2070_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_2071_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_2072_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_2073_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_2074_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_2075_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_2076_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_2077_not__exp__less__eq__0__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_2078_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2079_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2080_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2081_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2082_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2083_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_2084_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_2085_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_2086_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K3 )
               => ~ ( P @ I2 ) )
            & ( P @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_2087_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K3 )
          & ( ( plus_plus_nat @ I @ K3 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_2088_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_2089_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_2090_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_nat @ M @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2091_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N ) )
        = ( M = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2092_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( divide_divide_nat @ M @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2093_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_2094_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_2095_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_2096_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_2097_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( times_times_nat @ M @ N ) )
     => ( ( N = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2098_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_2099_mod__double__modulus,axiom,
    ! [M: code_integer,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
       => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X @ M ) )
          | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2100_mod__double__modulus,axiom,
    ! [M: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X @ M ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2101_mod__double__modulus,axiom,
    ! [M: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X @ M ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_2102_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2103_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2104_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_2105_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_2106_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_2107_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_2108_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_2109_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_2110_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_2111_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_2112_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_2113_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_2114_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_2115_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_2116_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_2117_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_2118_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_2119_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_2120_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_2121_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_2122_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_2123_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_2124_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_2125_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_2126_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_2127_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_2128_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_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_2136_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_2137_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_2138_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_2139_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_2140_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_2141_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_2142_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_2143_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_2144_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_2145_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).

% mod_mult2_eq
thf(fact_2146_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_2147_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_2148_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_2149_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_2150_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_2151_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_2152_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_2153_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_2154_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_2155_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_2156_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_2157_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_2158_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_2159_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_2160_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_2161_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_2162_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_2163_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_2164_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_2165_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_2166_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_2167_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_2168_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_2169_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_2170_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_2171_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_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_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_2180_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_2181_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_2182_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_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_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_2201_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_2202_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_2203_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_2204_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_2205_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_2206_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_2207_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_2208_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_2209_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_2210_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_2211_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_2212_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_2213_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_2214_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_2215_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_2216_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_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2228_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_2229_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2230_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2231_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2232_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2233_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2234_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2235_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2236_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2237_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_2238_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_2239_frac__less2,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2240_frac__less2,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_2241_frac__less,axiom,
    ! [X: real,Y: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2242_frac__less,axiom,
    ! [X: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_2243_frac__le,axiom,
    ! [Y: real,X: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2244_frac__le,axiom,
    ! [Y: rat,X: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_2245_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2246_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2247_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2248_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2249_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2250_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_2251_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_2252_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_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2260_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2261_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2262_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2263_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2264_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2265_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2266_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2267_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2268_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2269_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2270_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2271_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2272_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2273_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2274_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2275_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2276_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2277_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2278_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2279_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_2280_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_2281_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_2282_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_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2292_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2293_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2294_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2295_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_2296_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_2297_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_2298_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_2299_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_2300_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_2301_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_2302_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_2303_power__le__one,axiom,
    ! [A: real,N: 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 @ N ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_2304_power__le__one,axiom,
    ! [A: rat,N: 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 @ N ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2305_power__le__one,axiom,
    ! [A: nat,N: 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 @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_2306_power__le__one,axiom,
    ! [A: int,N: 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 @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2314_divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2315_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2316_add__divide__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2317_add__divide__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2318_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2319_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2320_add__num__frac,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2321_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_2322_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2323_add__frac__num,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2324_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_2325_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2326_add__frac__eq,axiom,
    ! [Y: real,Z: real,X: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2327_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M5: nat] :
      ( ( P @ X )
     => ( ! [X6: nat] :
            ( ( P @ X6 )
           => ( ord_less_eq_nat @ X6 @ M5 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_2339_length__pos__if__in__set,axiom,
    ! [X: complex,Xs: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2340_length__pos__if__in__set,axiom,
    ! [X: real,Xs: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2341_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2342_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2343_length__pos__if__in__set,axiom,
    ! [X: $o,Xs: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2344_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2345_length__pos__if__in__set,axiom,
    ! [X: int,Xs: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_2346_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2347_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ N @ M )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_2348_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2349_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( divide_divide_nat @ M @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2350_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
        = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2351_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2352_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N )
          = M )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2353_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_2354_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_2355_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_2356_power2__commute,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_2357_power2__commute,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_2358_power2__commute,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_2359_power2__commute,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_2360_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_2361_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_2362_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z5: real] :
          ( ( ord_less_real @ zero_zero_real @ Z5 )
         => ( ( ord_less_real @ Z5 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z5 @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2363_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z5 )
         => ( ( ord_less_rat @ Z5 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z5 @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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_2370_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_2371_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_2372_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_2373_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_2374_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_2375_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_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2391_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2392_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2393_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2413_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2414_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2415_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_2416_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_2417_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_2418_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_2419_power__Suc__less,axiom,
    ! [A: real,N: 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 @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2420_power__Suc__less,axiom,
    ! [A: rat,N: 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 @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2421_power__Suc__less,axiom,
    ! [A: nat,N: 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 @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2422_power__Suc__less,axiom,
    ! [A: int,N: 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 @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2423_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2424_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2425_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2426_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2427_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2428_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2429_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2430_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2431_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2432_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2433_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2434_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2435_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2436_self__le__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2437_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2438_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2439_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2440_one__less__power,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2441_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2442_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2443_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2444_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2445_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2446_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2447_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_2448_split__div,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I5: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I5 ) @ J3 ) )
               => ( P @ I5 ) ) ) ) ) ) ).

% split_div
thf(fact_2449_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_2450_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_2451_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_2452_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_2453_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_2454_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2455_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2456_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2457_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_2458_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_2459_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_2460_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_2461_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_2462_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_2463_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_2464_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2465_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2466_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2467_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2468_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2469_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2470_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2471_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_2472_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_2473_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_2474_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_2475_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2476_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2477_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2478_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2479_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2480_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( 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 @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2481_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_2482_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2483_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2484_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2485_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_2486_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2487_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2488_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2489_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2490_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2491_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2492_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2493_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2494_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2495_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2496_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2497_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2498_zero__le__even__power_H,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2499_zero__le__even__power_H,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2500_zero__le__even__power_H,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% zero_le_even_power'
thf(fact_2501_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_2502_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_2503_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_2504_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_2505_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X6 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X2 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ 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 @ N )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X6: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X6 )
                                      & ( ord_less_eq_nat @ X6 @ 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_2506_pred__list__to__short,axiom,
    ! [Deg: nat,X: 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 @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_2507_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: 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 @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_2508_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_2509_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_2510_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_2511_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_2512_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_2513_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_2514_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_2515_succ__min,axiom,
    ! [Deg: nat,X: 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 @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_2516_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( 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 @ X @ ( 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 ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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_2517_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y3: nat,X3: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).

% greater_shift
thf(fact_2518_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] : ( vEBT_VEBT_less @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ).

% less_shift
thf(fact_2519_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_2520_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_2521_idiff__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
      = N ) ).

% idiff_0_right
thf(fact_2522_idiff__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_2523_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_2524_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_2525_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_2526_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_2527_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_2528_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_2536_split__zdiv,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( divide_divide_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I5: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ I5 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I5: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ I5 ) ) ) ) ) ).

% split_zdiv
thf(fact_2537_split__zmod,axiom,
    ! [P: int > $o,N: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I5: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I5: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_2538_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_2539_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2540_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_2541_mod__pos__geq,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L2 )
     => ( ( ord_less_eq_int @ L2 @ K )
       => ( ( modulo_modulo_int @ K @ L2 )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).

% mod_pos_geq
thf(fact_2542_q__pos__lemma,axiom,
    ! [B4: int,Q4: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
     => ( ( ord_less_int @ R3 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).

% q_pos_lemma
thf(fact_2543_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_2544_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_2545_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_2546_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_2547_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_2548_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_2549_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_2550_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2551_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2552_div__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
      = ( ( K = zero_zero_int )
        | ( L2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_2553_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I5: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ I5 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_2554_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
        = ( ! [I5: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K @ I5 ) @ J3 ) ) )
             => ( P @ I5 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_2555_div__positive__int,axiom,
    ! [L2: int,K: int] :
      ( ( ord_less_eq_int @ L2 @ K )
     => ( ( ord_less_int @ zero_zero_int @ L2 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).

% div_positive_int
thf(fact_2556_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R: int,B4: int,Q4: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
       => ( ( ord_less_int @ R3 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_2557_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_2558_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_2559_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).

% int_div_less_self
thf(fact_2560_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_2561_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_2562_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_2563_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_2564_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_2565_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R: int,B4: int,Q4: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R3 ) @ zero_zero_int )
       => ( ( ord_less_int @ R @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_2566_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_int @ R @ B )
           => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_2567_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_2568_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_2569_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_2570_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_2571_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_2572_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_2573_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_2574_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B @ R )
         => ( ( ord_less_int @ B @ R3 )
           => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_2575_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_2576_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_2577_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_2578_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_2579_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_2580_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M
          = ( times_times_int @ D @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_2581_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q5: int] :
            ( M
            = ( times_times_int @ D @ Q5 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_2582_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_2583_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_2584_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_2585_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_2586_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_2587_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_2588_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_2589_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_2590_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ 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 ) @ X )
                          = ( 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_2591_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_2592_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_2593_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_2594_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( 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_2595_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_2596_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_2597_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_2598_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_2599_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_2600_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2601_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2602_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2603_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_2604_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_2605_mint__member,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_2606_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_2607_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_2608_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_2609_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_2610_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_2611_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_2612_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_2613_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_2614_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_2615_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = 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 ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_2616_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_2617_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y3: nat] :
            ( X
            = ( some_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_2618_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y3: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_2619_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y3: num] :
            ( X
            = ( some_num @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_2620_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y3: nat] :
            ( X
           != ( some_nat @ Y3 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_2621_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y3: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y3 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_2622_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y3: num] :
            ( X
           != ( some_num @ Y3 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_2623_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_2624_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_2625_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_2626_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_2627_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_2628_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_2636_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(3)
thf(fact_2637_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_2638_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_2639_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_2640_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_2641_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_2642_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(4)
thf(fact_2643_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_2644_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_2645_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_2646_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_2647_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_2648_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_2649_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_2650_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_2651_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_2652_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_2653_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_2654_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_2655_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_2656_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_2657_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_2658_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_2659_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_2660_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_2661_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_2662_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(5)
thf(fact_2663_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_2664_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_2665_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_2666_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_2667_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_2668_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(6)
thf(fact_2669_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_2670_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_2671_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_2672_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2673_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2674_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2675_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2676_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Z ) @ ( plus_p5714425477246183910nteger @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_2677_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2678_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2679_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2680_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2681_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( plus_p5714425477246183910nteger @ X @ ( ord_max_Code_integer @ Y @ Z ) )
      = ( ord_max_Code_integer @ ( plus_p5714425477246183910nteger @ X @ Y ) @ ( plus_p5714425477246183910nteger @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_2682_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ord_max_Code_integer @ ( minus_8373710615458151222nteger @ X @ Z ) @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2683_max__diff__distrib__left,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2684_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2685_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_2686_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_2687_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_2688_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_2689_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_2690_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
      = ( ord_max_nat @ N @ M ) ) ).

% nat_minus_add_max
thf(fact_2691_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_2692_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(3)
thf(fact_2693_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_2694_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_2695_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_2696_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_2697_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_2698_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_2699_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_2700_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_2701_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_2702_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_2703_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X3: nat] : ( P4 @ ( some_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_2704_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X3: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_2705_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X3: num] : ( P4 @ ( some_num @ X3 ) ) ) ) ) ).

% split_option_ex
thf(fact_2706_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X3: nat] : ( P4 @ ( some_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_2707_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X3: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_2708_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X3: num] : ( P4 @ ( some_num @ X3 ) ) ) ) ) ).

% split_option_all
thf(fact_2709_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2710_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2711_combine__options__cases,axiom,
    ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: nat,B5: num] :
              ( ( X
                = ( some_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2712_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2713_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2714_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2715_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2716_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2717_combine__options__cases,axiom,
    ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X @ Y ) )
       => ( ! [A5: num,B5: num] :
              ( ( X
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X @ Y ) ) )
         => ( P @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2718_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2719_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_2720_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_2721_times__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( times_times_int @ zero_zero_int @ L2 )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_2722_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_2723_plus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( plus_plus_int @ zero_zero_int @ L2 )
      = L2 ) ).

% plus_int_code(2)
thf(fact_2724_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_2725_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_2726_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_2727_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).

% int_distrib(1)
thf(fact_2728_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_2729_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_2730_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_2731_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_2732_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_2733_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_2734_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X6 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_2735_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_int @ I @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_2736_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_2737_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_2738_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_2739_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_2740_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_2741_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2742_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2743_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2744_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_2745_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_2746_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_2747_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_2748_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_2749_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_2750_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( N
          = ( 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_2751_aampt,axiom,
    ( ( ( xa = ma )
     => ( mi
        = ( if_nat
          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) )
            = 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 @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) ) ) ) ) )
    & ( ( xa != ma )
     => ( mi = ma ) ) ) ).

% aampt
thf(fact_2752_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2753_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).

% buildup_gives_valid
thf(fact_2754_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_2755_inrange,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( 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 ) ) @ N ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_2756_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_2757_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_2758_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_2759_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_2760_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] :
          ( X
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_2761_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_2762_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_2763_add__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_2764_mul__shift,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_2765_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_2766_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_2767_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_2768_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_2769_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_2770_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_2771_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_2772_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_2773_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_2774_atLeastAtMost__iff,axiom,
    ! [I: set_int,L2: set_int,U: set_int] :
      ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L2 @ U ) )
      = ( ( ord_less_eq_set_int @ L2 @ I )
        & ( ord_less_eq_set_int @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_2775_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_2776_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_2777_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_2778_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_2779_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_2780_Icc__eq__Icc,axiom,
    ! [L2: set_int,H2: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L2 @ H2 )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L2 = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_2781_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_2782_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_2783_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_2784_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_2785_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_2786_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_2787_add__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc_right
thf(fact_2788_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M @ N ) ) ).

% diff_Suc_Suc
thf(fact_2789_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_2790_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_2791_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_2792_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).

% max_Suc_Suc
thf(fact_2793_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2794_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2795_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2796_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2797_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2798_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2799_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2800_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2801_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2802_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_2803_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_2804_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_2805_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_2806_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_2807_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_2808_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_2809_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_2810_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2811_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( times_times_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2812_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_2813_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2814_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( ( power_power_nat @ X @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2815_mult__Suc__right,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ M @ ( suc @ N ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc_right
thf(fact_2816_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_2817_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_2818_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_2819_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_2820_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_2821_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_2822_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_2823_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_2824_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_2825_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_2826_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_2827_add__2__eq__Suc_H,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc'
thf(fact_2828_add__2__eq__Suc,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( suc @ ( suc @ N ) ) ) ).

% add_2_eq_Suc
thf(fact_2829_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_2830_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2831_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_2832_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_2833_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_2834_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_2835_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 ) @ X )
                  = ( 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 @ ( X = 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 @ ( X = 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_2836_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( 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 @ X @ ( 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 ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = 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_2837_del__x__not__mi,axiom,
    ! [Mi: nat,X: 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 @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X @ ( 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 @ X @ ( 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 ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = 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 ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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_2838_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ 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 ) @ X )
            = ( 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_2839_del__x__mi__lets__in__minNull,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ 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 ) @ X )
                            = ( 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_2840_del__x__mi__lets__in,axiom,
    ! [X: 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] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ 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 ) @ X )
                            = ( 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 ) @ X )
                            = ( 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_2841_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ 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 ) @ X )
                    = ( 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_2842_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ 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 ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = 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 ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = 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 ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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_2843_pred__less__length__list,axiom,
    ! [Deg: nat,X: 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 @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_2844_pred__lesseq__max,axiom,
    ! [Deg: nat,X: 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 @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_2845_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: 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 @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_2846_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: 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 @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_2847_dsimp,axiom,
    ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
    = ( vEBT_Node
      @ ( some_P7363390416028606310at_nat
        @ ( product_Pair_nat_nat @ mi
          @ ( if_nat @ ( xa = ma )
            @ ( if_nat
              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) )
                = 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 @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ) ) ) ) )
            @ ma ) ) )
      @ deg
      @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) )
      @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ xa @ na ) ) ) ) ).

% dsimp
thf(fact_2848_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X3: real] : ( times_times_real @ X3 @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_2849_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X3: rat] : ( times_times_rat @ X3 @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_2850_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X3: nat] : ( times_times_nat @ X3 @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_2851_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X3: int] : ( times_times_int @ X3 @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_2852_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_2853_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_2854_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2855_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A3: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2856_max__def__raw,axiom,
    ( ord_max_set_int
    = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2857_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2858_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2859_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2860_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def_raw
thf(fact_2861_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_2862_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2863_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2864_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2865_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2866_lambda__one,axiom,
    ( ( ^ [X3: complex] : X3 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_2867_lambda__one,axiom,
    ( ( ^ [X3: real] : X3 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_2868_lambda__one,axiom,
    ( ( ^ [X3: rat] : X3 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_2869_lambda__one,axiom,
    ( ( ^ [X3: nat] : X3 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_2870_lambda__one,axiom,
    ( ( ^ [X3: int] : X3 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_2871_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_2872_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_2873_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_2874_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_2875_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_2876_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_2877_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X6: nat] : ( P @ X6 @ zero_zero_nat )
     => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
       => ( ! [X6: nat,Y5: nat] :
              ( ( P @ X6 @ Y5 )
             => ( P @ ( suc @ X6 ) @ ( suc @ Y5 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_2878_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_2879_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_2880_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_2881_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_2882_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_2883_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_2884_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_2885_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_2886_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,K3: nat] :
              ( ( ord_less_nat @ I4 @ J2 )
             => ( ( ord_less_nat @ J2 @ K3 )
               => ( ( P @ I4 @ J2 )
                 => ( ( P @ J2 @ K3 )
                   => ( P @ I4 @ K3 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_2887_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_2888_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_2889_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_2890_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_2891_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
           => ( P @ I5 ) ) )
      = ( ( P @ N )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
           => ( P @ I5 ) ) ) ) ).

% All_less_Suc
thf(fact_2892_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_2893_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_2894_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
            & ( P @ I5 ) ) )
      = ( ( P @ N )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
            & ( P @ I5 ) ) ) ) ).

% Ex_less_Suc
thf(fact_2895_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_2896_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_2897_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_2898_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_2899_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_2900_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_2901_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_2902_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R4: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X6: nat] : ( R4 @ X6 @ X6 )
       => ( ! [X6: nat,Y5: nat,Z5: nat] :
              ( ( R4 @ X6 @ Y5 )
             => ( ( R4 @ Y5 @ Z5 )
               => ( R4 @ X6 @ Z5 ) ) )
         => ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
           => ( R4 @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_2903_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_2904_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_2905_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_2906_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_2907_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_2908_Suc__le__D,axiom,
    ! [N: nat,M7: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
     => ? [M4: nat] :
          ( M7
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_2909_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_2910_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_2911_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_2912_add__Suc__shift,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_2913_add__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N )
      = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).

% add_Suc
thf(fact_2914_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_2915_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_2916_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( M = N ) ) ).

% Suc_mult_cancel1
thf(fact_2917_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_2918_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_code(2)
thf(fact_2919_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_code(2)
thf(fact_2920_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_code(2)
thf(fact_2921_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_code(2)
thf(fact_2922_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).

% mod_Suc_eq
thf(fact_2923_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_2924_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).

% set_bit_greater_eq
thf(fact_2925_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_2926_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_2927_power__numeral__even,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2928_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_2929_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_2930_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2931_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_2932_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_2933_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_2934_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_2935_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_2936_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less
thf(fact_2937_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less
thf(fact_2938_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_2939_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_2940_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_2941_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_2942_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_2943_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_2944_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_2945_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_2946_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_2947_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N5 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_2948_power__Suc,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).

% power_Suc
thf(fact_2949_power__Suc,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).

% power_Suc
thf(fact_2950_power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).

% power_Suc
thf(fact_2951_power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_Suc
thf(fact_2952_power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).

% power_Suc
thf(fact_2953_power__Suc2,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_2954_power__Suc2,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_2955_power__Suc2,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_2956_power__Suc2,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_2957_power__Suc2,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).

% power_Suc2
thf(fact_2958_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2959_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2960_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2961_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2962_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_2963_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2964_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2965_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2966_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2967_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_2968_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_2969_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_2970_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
           => ( P @ I5 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
           => ( P @ ( suc @ I5 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_2971_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_2972_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( suc @ N ) )
            & ( P @ I5 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ N )
            & ( P @ ( suc @ I5 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_2973_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_2974_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_2975_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_2976_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_2977_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_2978_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_2979_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_2980_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_2981_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_2982_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_2983_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_2984_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2985_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_2986_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2987_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_2988_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_2989_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_2990_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N2: nat] :
        ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_2991_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_2992_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2993_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2994_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2995_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_2996_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_2997_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_2998_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_2999_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_3000_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_3001_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_3002_mult__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).

% mult_Suc
thf(fact_3003_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_3004_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_3005_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_3006_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_3007_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_3008_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P5: nat,M: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P5 )
       => ( ( ord_less_nat @ M @ P5 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P5 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P5 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_3009_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_3010_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3011_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3012_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3013_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3014_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3015_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3016_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3017_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3018_power__gt1,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3019_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3020_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3021_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_3022_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_3023_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K3 )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K3 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_3024_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_3025_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_3026_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_3027_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_3028_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3029_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3030_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3031_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3032_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_3033_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3034_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3035_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3036_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3037_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_3038_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_3039_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3040_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3041_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).

% one_less_mult
thf(fact_3042_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_3043_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).

% power_gt_expt
thf(fact_3044_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R2: real] :
          ( ( ord_less_real @ zero_zero_real @ R2 )
          & ( ( power_power_real @ R2 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3045_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).

% nat_one_le_power
thf(fact_3046_verit__la__generic,axiom,
    ! [A: int,X: int] :
      ( ( ord_less_eq_int @ A @ X )
      | ( A = X )
      | ( ord_less_eq_int @ X @ A ) ) ).

% verit_la_generic
thf(fact_3047_power__Suc__le__self,axiom,
    ! [A: real,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3048_power__Suc__le__self,axiom,
    ! [A: rat,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3049_power__Suc__le__self,axiom,
    ! [A: nat,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3050_power__Suc__le__self,axiom,
    ! [A: int,N: 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 @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_3051_power__Suc__less__one,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_3052_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_3053_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_3054_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_3055_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3056_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_3057_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_3058_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N2 )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).

% div_if
thf(fact_3059_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_3060_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3061_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_3062_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_3063_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).

% add_eq_if
thf(fact_3064_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_3065_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3066_set__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3067_set__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_3068_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_3069_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_3070_less__2__cases,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N = zero_zero_nat )
        | ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_3071_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( ( divide_divide_nat @ M @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_3072_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
            & ( P @ Q5 ) ) ) ) ).

% split_div'
thf(fact_3073_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_3074_power__odd__eq,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3075_power__odd__eq,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3076_power__odd__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3077_power__odd__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3078_power__odd__eq,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3079_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P @ N3 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_3080_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_3081_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_3082_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
      = ( ord_less_real @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3083_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ( ~ ( ord_less_eq_rat @ B4 @ A4 ) )
      = ( ord_less_rat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3084_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
      = ( ord_less_num @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3085_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
      = ( ord_less_nat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3086_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
      = ( ord_less_int @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_3087_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_3088_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_3089_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_3090_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_3091_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_3092_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_3093_odd__0__le__power__imp__0__le,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3094_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3095_odd__0__le__power__imp__0__le,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3096_odd__power__less__zero,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_3097_odd__power__less__zero,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3098_odd__power__less__zero,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_3099_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_3100_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X6 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X6: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_3101_div__less__mono,axiom,
    ! [A2: nat,B3: nat,N: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A2 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B3 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3102_div__mod__decomp,axiom,
    ! [A2: nat,N: nat] :
      ( A2
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).

% div_mod_decomp
thf(fact_3103_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ( ( modulo_modulo_int @ A2 @ N )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B3 @ N ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_3104_div__mod__decomp__int,axiom,
    ! [A2: int,N: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_3105_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X6 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X2 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X6 @ 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 @ N )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X6: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X6 @ N )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X6 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X6 )
                                      & ( ord_less_eq_nat @ X6 @ 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_3106_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = 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 ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = 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 ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_3107_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_3108_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_3109_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3110_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_3111_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_3112_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_3113_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_3114_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_3115_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_3116_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_3117_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_3118_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).

% vebt_member.simps(2)
thf(fact_3119_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_3120_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs2: set_nat,X3: nat,Y3: nat] :
          ( ( member_nat @ Y3 @ Xs2 )
          & ( ord_less_nat @ Y3 @ X3 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs2 )
             => ( ( ord_less_nat @ Z3 @ X3 )
               => ( ord_less_eq_nat @ Z3 @ Y3 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_3121_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs2: set_nat,X3: nat,Y3: nat] :
          ( ( member_nat @ Y3 @ Xs2 )
          & ( ord_less_nat @ X3 @ Y3 )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs2 )
             => ( ( ord_less_nat @ X3 @ Z3 )
               => ( ord_less_eq_nat @ Y3 @ Z3 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_3122_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_3123_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_3124_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_3125_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_3126_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_3127_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_3128_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_3129_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_3130_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A5: $o] :
              ( ? [Uw2: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [Va2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va2 ) ) )
                 => ~ ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( ( A5
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A5
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3131_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N3: nat] :
                  ( Xa2
                  = ( suc @ N3 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3132_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B5 ) ) ) )
       => ( ! [A5: $o] :
              ( ? [B5: $o] :
                  ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A5 @ $false ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ? [N3: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N3 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A5 @ B5 ) ) ) )
           => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( 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 ) )
                             => ( Y
                                = ( 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 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( 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_3133_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( P @ X6 )
              = ( P6 @ X6 ) ) )
       => ( ! [X6: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ A2 )
                     => ( X6
                       != ( minus_minus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X6 )
               => ( P @ ( plus_plus_int @ X6 @ D4 ) ) ) )
         => ( ! [X6: int,K3: int] :
                ( ( P6 @ X6 )
                = ( P6 @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X2: int] : ( P @ X2 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ A2 )
                        & ( P @ ( minus_minus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3134_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( P @ X6 )
              = ( P6 @ X6 ) ) )
       => ( ! [X6: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb: int] :
                      ( ( member_int @ Xb @ B3 )
                     => ( X6
                       != ( plus_plus_int @ Xb @ Xa ) ) ) )
             => ( ( P @ X6 )
               => ( P @ ( minus_minus_int @ X6 @ D4 ) ) ) )
         => ( ! [X6: int,K3: int] :
                ( ( P6 @ X6 )
                = ( P6 @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D4 ) ) ) )
           => ( ( ? [X2: int] : ( P @ X2 ) )
              = ( ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X3 ) )
                | ? [X3: int] :
                    ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y3: int] :
                        ( ( member_int @ Y3 @ B3 )
                        & ( P @ ( plus_plus_int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3135_bset_I6_J,axiom,
    ! [D4: int,B3: 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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_3136_bset_I8_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X4 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3137_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_3138_deg__1__Leafy,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( N = one_one_nat )
       => ? [A5: $o,B5: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).

% deg_1_Leafy
thf(fact_3139_deg__1__Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
     => ? [A5: $o,B5: $o] :
          ( T
          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ).

% deg_1_Leaf
thf(fact_3140_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B2: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_3141_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_3142_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3143_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_3144_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3145_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_3146_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_3147_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_3148_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_3149_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_3150_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3151_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X6: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X6 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X6: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X6 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3152_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_3153_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_3154_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_3155_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_3156_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3157_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_3158_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3159_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X6: real] :
          ( ( ord_less_real @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3160_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X6: rat] :
          ( ( ord_less_rat @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3161_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X6: num] :
          ( ( ord_less_num @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3162_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X6: nat] :
          ( ( ord_less_nat @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3163_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X6: int] :
          ( ( ord_less_int @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3164_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X6: real] :
          ( ( ord_less_real @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3165_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X6: rat] :
          ( ( ord_less_rat @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3166_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X6: num] :
          ( ( ord_less_num @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3167_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X6: nat] :
          ( ( ord_less_nat @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3168_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X6: int] :
          ( ( ord_less_int @ Z4 @ X6 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ Z4 @ X6 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z5 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3169_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3170_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3171_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3172_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3173_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_3174_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3175_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3176_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3177_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3178_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_3179_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ~ ( ord_less_real @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3180_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3181_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ~ ( ord_less_num @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3182_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3183_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ~ ( ord_less_int @ X4 @ T ) ) ).

% pinf(5)
thf(fact_3184_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( ord_less_real @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3185_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( ord_less_rat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3186_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ( ord_less_num @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3187_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3188_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( ord_less_int @ T @ X4 ) ) ).

% pinf(7)
thf(fact_3189_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X6: real] :
          ( ( ord_less_real @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3190_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X6: rat] :
          ( ( ord_less_rat @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3191_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X6: num] :
          ( ( ord_less_num @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3192_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X6: nat] :
          ( ( ord_less_nat @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3193_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X6: int] :
          ( ( ord_less_int @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3194_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z4: real] :
        ! [X6: real] :
          ( ( ord_less_real @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: real] :
          ! [X6: real] :
            ( ( ord_less_real @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3195_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z4: rat] :
        ! [X6: rat] :
          ( ( ord_less_rat @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: rat] :
          ! [X6: rat] :
            ( ( ord_less_rat @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3196_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z4: num] :
        ! [X6: num] :
          ( ( ord_less_num @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: num] :
          ! [X6: num] :
            ( ( ord_less_num @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3197_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z4: nat] :
        ! [X6: nat] :
          ( ( ord_less_nat @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: nat] :
          ! [X6: nat] :
            ( ( ord_less_nat @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3198_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z4: int] :
        ! [X6: int] :
          ( ( ord_less_int @ X6 @ Z4 )
         => ( ( P @ X6 )
            = ( P6 @ X6 ) ) )
     => ( ? [Z4: int] :
          ! [X6: int] :
            ( ( ord_less_int @ X6 @ Z4 )
           => ( ( Q @ X6 )
              = ( Q6 @ X6 ) ) )
       => ? [Z5: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z5 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3199_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3200_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3201_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3202_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3203_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_3204_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3205_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3206_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3207_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3208_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_3209_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( ord_less_real @ X4 @ T ) ) ).

% minf(5)
thf(fact_3210_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( ord_less_rat @ X4 @ T ) ) ).

% minf(5)
thf(fact_3211_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ( ord_less_num @ X4 @ T ) ) ).

% minf(5)
thf(fact_3212_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_3213_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( ord_less_int @ X4 @ T ) ) ).

% minf(5)
thf(fact_3214_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ~ ( ord_less_real @ T @ X4 ) ) ).

% minf(7)
thf(fact_3215_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ~ ( ord_less_rat @ T @ X4 ) ) ).

% minf(7)
thf(fact_3216_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ~ ( ord_less_num @ T @ X4 ) ) ).

% minf(7)
thf(fact_3217_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_3218_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ~ ( ord_less_int @ T @ X4 ) ) ).

% minf(7)
thf(fact_3219_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3220_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_3221_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3222_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3223_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_3224_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_3225_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X6: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X6 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X6: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X6 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X6: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X6 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3226_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,Uw2: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va2 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_3227_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_3228_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) )
     => ( ! [A5: $o,B5: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A5: $o,B5: $o,N3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N3 ) ) ) )
         => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X6: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X6 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X6: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X6 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_3229_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X6: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X6 ) )
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X6: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ X6 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X6: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X6 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_3230_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X6: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X6 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X6: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X6 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X6: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X6 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X6: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X6 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X6: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X6 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_3231_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_3232_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ~ ( ord_less_eq_real @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3233_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3234_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3235_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3236_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T ) ) ).

% pinf(6)
thf(fact_3237_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( ord_less_eq_real @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3238_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( ord_less_eq_rat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3239_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z5 @ X4 )
     => ( ord_less_eq_num @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3240_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3241_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( ord_less_eq_int @ T @ X4 ) ) ).

% pinf(8)
thf(fact_3242_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( ord_less_eq_real @ X4 @ T ) ) ).

% minf(6)
thf(fact_3243_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( ord_less_eq_rat @ X4 @ T ) ) ).

% minf(6)
thf(fact_3244_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ( ord_less_eq_num @ X4 @ T ) ) ).

% minf(6)
thf(fact_3245_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_3246_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( ord_less_eq_int @ X4 @ T ) ) ).

% minf(6)
thf(fact_3247_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ~ ( ord_less_eq_real @ T @ X4 ) ) ).

% minf(8)
thf(fact_3248_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).

% minf(8)
thf(fact_3249_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z5: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z5 )
     => ~ ( ord_less_eq_num @ T @ X4 ) ) ).

% minf(8)
thf(fact_3250_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_3251_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ~ ( ord_less_eq_int @ T @ X4 ) ) ).

% minf(8)
thf(fact_3252_inf__period_I1_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X6: real,K3: real] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X6: real,K3: real] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ 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_3253_inf__period_I1_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X6: rat,K3: rat] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X6: rat,K3: rat] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X4: rat,K4: rat] :
            ( ( ( P @ X4 )
              & ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) )
              & ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_3254_inf__period_I1_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X6: int,K3: int] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X6: int,K3: int] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ 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_3255_inf__period_I2_J,axiom,
    ! [P: real > $o,D4: real,Q: real > $o] :
      ( ! [X6: real,K3: real] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ D4 ) ) ) )
     => ( ! [X6: real,K3: real] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_real @ X6 @ ( times_times_real @ K3 @ 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_3256_inf__period_I2_J,axiom,
    ! [P: rat > $o,D4: rat,Q: rat > $o] :
      ( ! [X6: rat,K3: rat] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K3 @ D4 ) ) ) )
     => ( ! [X6: rat,K3: rat] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_rat @ X6 @ ( times_times_rat @ K3 @ D4 ) ) ) )
       => ! [X4: rat,K4: rat] :
            ( ( ( P @ X4 )
              | ( Q @ X4 ) )
            = ( ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) )
              | ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_3257_inf__period_I2_J,axiom,
    ! [P: int > $o,D4: int,Q: int > $o] :
      ( ! [X6: int,K3: int] :
          ( ( P @ X6 )
          = ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D4 ) ) ) )
     => ( ! [X6: int,K3: int] :
            ( ( Q @ X6 )
            = ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ 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_3258_conj__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P6: $o] :
      ( ( X = X5 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_3259_imp__le__cong,axiom,
    ! [X: int,X5: int,P: $o,P6: $o] :
      ( ( X = X5 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X5 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_3260_bset_I1_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X6
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X6 )
           => ( P @ ( minus_minus_int @ X6 @ D4 ) ) ) )
     => ( ! [X6: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X6
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X6 )
             => ( Q @ ( minus_minus_int @ X6 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_3261_bset_I2_J,axiom,
    ! [D4: int,B3: set_int,P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ B3 )
                 => ( X6
                   != ( plus_plus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X6 )
           => ( P @ ( minus_minus_int @ X6 @ D4 ) ) ) )
     => ( ! [X6: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ B3 )
                   => ( X6
                     != ( plus_plus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X6 )
             => ( Q @ ( minus_minus_int @ X6 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_3262_aset_I1_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X6 )
           => ( P @ ( plus_plus_int @ X6 @ D4 ) ) ) )
     => ( ! [X6: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X6 )
             => ( Q @ ( plus_plus_int @ X6 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_3263_aset_I2_J,axiom,
    ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ! [Xa: int] :
              ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb: int] :
                  ( ( member_int @ Xb @ A2 )
                 => ( X6
                   != ( minus_minus_int @ Xb @ Xa ) ) ) )
         => ( ( P @ X6 )
           => ( P @ ( plus_plus_int @ X6 @ D4 ) ) ) )
     => ( ! [X6: int] :
            ( ! [Xa: int] :
                ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb: int] :
                    ( ( member_int @ Xb @ A2 )
                   => ( X6
                     != ( minus_minus_int @ Xb @ Xa ) ) ) )
           => ( ( Q @ X6 )
             => ( Q @ ( plus_plus_int @ X6 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_3264_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X6: int,K3: int] :
            ( ( P6 @ X6 )
            = ( P6 @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X6: int] :
              ( ( ord_less_int @ Z4 @ X6 )
             => ( ( P @ X6 )
                = ( P6 @ X6 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3265_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X6: int,K3: int] :
            ( ( P1 @ X6 )
            = ( P1 @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X6: int] :
              ( ( ord_less_int @ X6 @ Z4 )
             => ( ( P @ X6 )
                = ( P1 @ X6 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3266_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( 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_3267_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( 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_3268_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3269_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X6: int] :
            ( ( P @ X6 )
           => ( P @ ( plus_plus_int @ X6 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3270_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X6: int] :
            ( ( P @ X6 )
           => ( P @ ( minus_minus_int @ X6 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3271_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X6: int,K3: int] :
            ( ( P @ X6 )
            = ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) )
       => ( ( ? [X2: int] : ( P @ X2 ) )
          = ( ? [X3: int] :
                ( ( member_int @ X3 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X3 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3272_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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X4 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_3273_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 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X4 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_3274_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 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( plus_plus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_3275_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 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( plus_plus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_3276_bset_I7_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X4 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3277_bset_I5_J,axiom,
    ! [D4: int,B3: 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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X4 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_3278_bset_I4_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 != T )
             => ( ( minus_minus_int @ X4 @ D4 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_3279_bset_I3_J,axiom,
    ! [D4: int,T: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
       => ! [X4: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X4 = T )
             => ( ( minus_minus_int @ X4 @ D4 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_3280_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A3: $o,B2: $o] :
                ( A1
                = ( vEBT_Leaf @ A3 @ B2 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X2 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X2 )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ N2 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ N2 ) )
              & ! [I5: nat] :
                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                 => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X2 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I5 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I5 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                              = I5 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X3 @ N2 ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
              & ( A22
                = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
              & ! [I5: nat] :
                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                 => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ X2 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I5 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
                          = I5 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
                      & ! [X3: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                              = I5 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I5 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3281_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A5: $o,B5: $o] :
              ( A12
              = ( vEBT_Leaf @ A5 @ B5 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: 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 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M4 ) )
                         => ( ~ ? [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,N3: nat,Summary2: vEBT_VEBT,M4: 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 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M4 ) )
                           => ( ~ ? [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,N3: nat,Summary2: vEBT_VEBT,M4: 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 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M4 ) )
                             => ( ! [I2: nat] :
                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X2 ) )
                                      = ( 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 ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I2 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X4: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                                        = I2 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X4 )
                                                      & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: 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 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M4 ) )
                               => ( ! [I2: nat] :
                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X2 ) )
                                        = ( 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 ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I2 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X4: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                                                          = I2 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X4 )
                                                        & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3282_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B5 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3283_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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X4 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_3284_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 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_3285_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( A5
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A5
                 => ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3286_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( B5
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B5
                 => ( ( A5
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A5
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3287_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_3288_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_3289_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3290_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( 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] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3291_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3292_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A5 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B5 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( 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_3293_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_3294_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_3295_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X3: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X3 )
          | ( vEBT_VEBT_membermima @ T2 @ X3 ) ) ) ) ).

% both_member_options_def
thf(fact_3296_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_3297_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_3298_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_3299_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3300_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_3301_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A5: nat,B5: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A5 ) @ ( some_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3302_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A5 ) @ ( some_P7363390416028606310at_nat @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3303_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A5: num,B5: num] :
              ( X
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A5 ) @ ( some_num @ B5 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_3304_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X6: nat,Y5: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X6 ) @ ( some_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3305_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X6: product_prod_nat_nat,Y5: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X6 ) @ ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3306_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X6: num,Y5: num] :
              ( X
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X6 ) @ ( some_num @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_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_3314_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_3315_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_3316_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb3: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb3 )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb3 = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A5: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A5 ) )
               => ! [B5: product_prod_nat_nat] :
                    ( ( Xb3
                      = ( some_P7363390416028606310at_nat @ B5 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3317_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa2: option_num,Xb3: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb3 )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb3 = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A5: num] :
                ( ( Xa2
                  = ( some_num @ A5 ) )
               => ! [B5: num] :
                    ( ( Xb3
                      = ( some_num @ B5 ) )
                   => ( Y
                     != ( some_num @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3318_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: option_nat,Xb3: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb3 )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb3 = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A5: nat] :
                ( ( Xa2
                  = ( some_nat @ A5 ) )
               => ! [B5: nat] :
                    ( ( Xb3
                      = ( some_nat @ B5 ) )
                   => ( Y
                     != ( some_nat @ ( X @ A5 @ B5 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_3319_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_3320_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_3321_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_3322_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_3323_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_3324_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A5: $o,B5: $o] :
          ( X
         != ( vEBT_Leaf @ A5 @ B5 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_3325_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_3326_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_3327_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( 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] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_3328_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A5: $o,B5: $o] :
            ( ( X
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A5 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B5 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( 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_3329_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y3: nat] :
                ( ( vEBT_vebt_member @ T @ Y3 )
                & ( ord_less_nat @ X @ Y3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_3330_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y3: nat] :
                ( ( vEBT_vebt_member @ T @ Y3 )
                & ( ord_less_nat @ Y3 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_3331_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B5
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B5
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N3: nat] :
                    ( ( Xa2
                      = ( suc @ N3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3332_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_set_int @ A2 @ B3 ) ) ) ).

% psubsetI
thf(fact_3333_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,Uw2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ Uw2 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [Va2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va2 ) ) )
                     => ( ( ( B5
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B5
                           => ( ( A5
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A5
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
                   => ( ( Y = 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,Ve2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3334_max_Oabsorb3,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3335_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3336_max_Oabsorb3,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_max_real @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3337_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3338_max_Oabsorb3,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3339_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3340_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_3341_max_Oabsorb4,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3342_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3343_max_Oabsorb4,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_max_real @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3344_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3345_max_Oabsorb4,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3346_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3347_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_3348_max__less__iff__conj,axiom,
    ! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X @ Z )
        & ( ord_le72135733267957522d_enat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3349_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X @ Y ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X @ Z )
        & ( ord_le6747313008572928689nteger @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3350_max__less__iff__conj,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
      = ( ( ord_less_real @ X @ Z )
        & ( ord_less_real @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3351_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
      = ( ( ord_less_rat @ X @ Z )
        & ( ord_less_rat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3352_max__less__iff__conj,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
      = ( ( ord_less_num @ X @ Z )
        & ( ord_less_num @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3353_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3354_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3355_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_3356_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_3357_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_3358_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_3359_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_3360_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_3361_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_3362_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_3363_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_3364_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_3365_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% subset_antisym
thf(fact_3366_subsetI,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( member_complex @ X6 @ B3 ) )
     => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).

% subsetI
thf(fact_3367_subsetI,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( member_real @ X6 @ B3 ) )
     => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).

% subsetI
thf(fact_3368_subsetI,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ! [X6: set_nat] :
          ( ( member_set_nat @ X6 @ A2 )
         => ( member_set_nat @ X6 @ B3 ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).

% subsetI
thf(fact_3369_subsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( member_nat @ X6 @ B3 ) )
     => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).

% subsetI
thf(fact_3370_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( member_int @ X6 @ B3 ) )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% subsetI
thf(fact_3371_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_3372_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_3373_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_3374_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_3375_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_3376_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_3377_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_3378_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_3379_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_3380_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_3381_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_3382_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_3383_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_3384_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_3385_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_3386_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_3387_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_3388_max_Oabsorb2,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ B )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3389_max_Oabsorb2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_max_Code_integer @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3390_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3391_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3392_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3393_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_3394_max_Oabsorb1,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( ( ord_ma741700101516333627d_enat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3395_max_Oabsorb1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( ( ord_max_Code_integer @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3396_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3397_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3398_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3399_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_3400_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3401_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_3402_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_3403_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_3404_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_3405_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_3406_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3407_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_3408_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_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_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_3416_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_3417_double__diff,axiom,
    ! [A2: set_nat,B3: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C4 )
       => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3418_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_3419_Diff__subset,axiom,
    ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).

% Diff_subset
thf(fact_3420_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).

% Diff_subset
thf(fact_3421_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D4 @ B3 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3422_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D4: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D4 @ B3 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_3423_Collect__mono__iff,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
      = ( ! [X3: product_prod_int_int] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3424_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X3: complex] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3425_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X3: set_nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3426_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3427_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X3: int] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_3428_set__eq__subset,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_3429_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_3430_Collect__mono,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X6: product_prod_int_int] :
          ( ( P @ X6 )
         => ( Q @ X6 ) )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3431_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X6: complex] :
          ( ( P @ X6 )
         => ( Q @ X6 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_3432_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X6: set_nat] :
          ( ( P @ X6 )
         => ( Q @ X6 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3433_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X6: nat] :
          ( ( P @ X6 )
         => ( Q @ X6 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_3434_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X6: int] :
          ( ( P @ X6 )
         => ( Q @ X6 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_3435_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_3436_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A6 )
         => ( member_complex @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3437_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A6 )
         => ( member_real @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3438_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A6 )
         => ( member_set_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3439_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A6 )
         => ( member_nat @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3440_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A6 )
         => ( member_int @ T2 @ B6 ) ) ) ) ).

% subset_iff
thf(fact_3441_equalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% equalityD2
thf(fact_3442_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% equalityD1
thf(fact_3443_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [X3: complex] :
          ( ( member_complex @ X3 @ A6 )
         => ( member_complex @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3444_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X3: real] :
          ( ( member_real @ X3 @ A6 )
         => ( member_real @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3445_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
        ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A6 )
         => ( member_set_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3446_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
         => ( member_nat @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3447_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X3: int] :
          ( ( member_int @ X3 @ A6 )
         => ( member_int @ X3 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_3448_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% equalityE
thf(fact_3449_subsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B3 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B3 ) ) ) ).

% subsetD
thf(fact_3450_subsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B3 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B3 ) ) ) ).

% subsetD
thf(fact_3451_subsetD,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_3452_subsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_3453_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B3 ) ) ) ).

% subsetD
thf(fact_3454_in__mono,axiom,
    ! [A2: set_complex,B3: set_complex,X: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B3 )
     => ( ( member_complex @ X @ A2 )
       => ( member_complex @ X @ B3 ) ) ) ).

% in_mono
thf(fact_3455_in__mono,axiom,
    ! [A2: set_real,B3: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ B3 )
     => ( ( member_real @ X @ A2 )
       => ( member_real @ X @ B3 ) ) ) ).

% in_mono
thf(fact_3456_in__mono,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
     => ( ( member_set_nat @ X @ A2 )
       => ( member_set_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_3457_in__mono,axiom,
    ! [A2: set_nat,B3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( member_nat @ X @ A2 )
       => ( member_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_3458_in__mono,axiom,
    ! [A2: set_int,B3: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ X @ A2 )
       => ( member_int @ X @ B3 ) ) ) ).

% in_mono
thf(fact_3459_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ( ord_less_set_complex @ A2 @ B3 )
     => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3460_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( ord_less_set_real @ A2 @ B3 )
     => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3461_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B3: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B3 )
     => ? [B5: set_nat] : ( member_set_nat @ B5 @ ( minus_2163939370556025621et_nat @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3462_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3463_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B3 )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3464_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X3: real] :
            ( ( member_real @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3465_Collect__subset,axiom,
    ! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
      ( ord_le2843351958646193337nt_int
      @ ( collec213857154873943460nt_int
        @ ^ [X3: product_prod_int_int] :
            ( ( member5262025264175285858nt_int @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3466_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X3: complex] :
            ( ( member_complex @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3467_Collect__subset,axiom,
    ! [A2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3468_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3469_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X3: int] :
            ( ( member_int @ X3 @ A2 )
            & ( P @ X3 ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_3470_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
          @ ^ [X3: complex] : ( member_complex @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3471_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X3: real] : ( member_real @ X3 @ A6 )
          @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3472_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
          @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3473_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3474_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X3: int] : ( member_int @ X3 @ A6 )
          @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_3475_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_3476_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_3477_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_3478_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_3479_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_3480_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_3481_max_OorderE,axiom,
    ! [B: extended_enat,A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B @ A )
     => ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3482_max_OorderE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B @ A )
     => ( A
        = ( ord_max_Code_integer @ A @ B ) ) ) ).

% max.orderE
thf(fact_3483_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3484_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_3485_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_3486_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_3487_max_OorderI,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( A
        = ( ord_ma741700101516333627d_enat @ A @ B ) )
     => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).

% max.orderI
thf(fact_3488_max_OorderI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( ord_max_Code_integer @ A @ B ) )
     => ( ord_le3102999989581377725nteger @ B @ A ) ) ).

% max.orderI
thf(fact_3489_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_3490_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_3491_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_3492_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_3493_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_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_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_3504_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_3505_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( A3
          = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3506_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A3: code_integer] :
          ( A3
          = ( ord_max_Code_integer @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3507_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( A3
          = ( ord_max_rat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3508_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( A3
          = ( ord_max_num @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3509_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( A3
          = ( ord_max_nat @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3510_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( A3
          = ( ord_max_int @ A3 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_3511_max_Ocobounded1,axiom,
    ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3512_max_Ocobounded1,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded1
thf(fact_3513_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3514_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_3515_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3516_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_3517_max_Ocobounded2,axiom,
    ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3518_max_Ocobounded2,axiom,
    ! [B: code_integer,A: code_integer] : ( ord_le3102999989581377725nteger @ B @ ( ord_max_Code_integer @ A @ B ) ) ).

% max.cobounded2
thf(fact_3519_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3520_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_3521_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3522_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_3523_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X )
        | ( ord_le2932123472753598470d_enat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3524_le__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le3102999989581377725nteger @ Z @ X )
        | ( ord_le3102999989581377725nteger @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3525_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z @ X )
        | ( ord_less_eq_rat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3526_le__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z @ X )
        | ( ord_less_eq_num @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3527_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3528_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3529_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3530_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A3: code_integer] :
          ( ( ord_max_Code_integer @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3531_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_max_rat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3532_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_max_num @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3533_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_max_nat @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3534_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_max_int @ A3 @ B2 )
          = A3 ) ) ) ).

% max.absorb_iff1
thf(fact_3535_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3536_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( ord_max_Code_integer @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3537_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_max_rat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3538_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_max_num @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3539_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_max_nat @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3540_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_max_int @ A3 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_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_3552_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_3553_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X: extended_enat,Y: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X )
        | ( ord_le72135733267957522d_enat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3554_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X @ Y ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X )
        | ( ord_le6747313008572928689nteger @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3555_less__max__iff__disj,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
      = ( ( ord_less_real @ Z @ X )
        | ( ord_less_real @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3556_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Z @ X )
        | ( ord_less_rat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3557_less__max__iff__disj,axiom,
    ! [Z: num,X: num,Y: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_num @ Z @ X )
        | ( ord_less_num @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3558_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3559_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_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_3567_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A3: extended_enat] :
          ( ( A3
            = ( ord_ma741700101516333627d_enat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3568_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A3: code_integer] :
          ( ( A3
            = ( ord_max_Code_integer @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3569_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( A3
            = ( ord_max_real @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3570_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( A3
            = ( ord_max_rat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3571_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( A3
            = ( ord_max_num @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3572_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( A3
            = ( ord_max_nat @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3573_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( A3
            = ( ord_max_int @ A3 @ B2 ) )
          & ( A3 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3574_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_3575_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_3576_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_3577_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_3578_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_3579_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_3580_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_3581_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_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% psubsetE
thf(fact_3589_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_3590_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_3591_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_3592_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_3593_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_3594_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_3595_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B5 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A5: $o,B5: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A5 @ B5 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A5 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A5: $o,B5: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ A5 @ B5 ) )
                 => ! [N3: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N3 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( 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,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( 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 ) )
                               => ( Y
                                  = ( 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 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( 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_3596_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B5 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3597_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( 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_3598_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ 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_3599_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A5 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B5 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ 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_3600_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ 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_3601_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3602_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa2 ) )
               => ~ ( ( ( Xa2 = zero_zero_nat )
                     => A5 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B5 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( 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_3603_DiffI,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( ( member_complex @ C @ A2 )
     => ( ~ ( member_complex @ C @ B3 )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_3604_DiffI,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B3 )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_3605_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ~ ( member_set_nat @ C @ B3 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_3606_DiffI,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B3 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_3607_DiffI,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).

% DiffI
thf(fact_3608_Diff__iff,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
      = ( ( member_complex @ C @ A2 )
        & ~ ( member_complex @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_3609_Diff__iff,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_3610_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
      = ( ( member_set_nat @ C @ A2 )
        & ~ ( member_set_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_3611_Diff__iff,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_3612_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_3613_Diff__idemp,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
      = ( minus_minus_set_nat @ A2 @ B3 ) ) ).

% Diff_idemp
thf(fact_3614_DiffE,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
     => ~ ( ( member_complex @ C @ A2 )
         => ( member_complex @ C @ B3 ) ) ) ).

% DiffE
thf(fact_3615_DiffE,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B3 ) ) ) ).

% DiffE
thf(fact_3616_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ( member_set_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_3617_DiffE,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B3 ) ) ) ).

% DiffE
thf(fact_3618_DiffE,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_3619_DiffD1,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
     => ( member_complex @ C @ A2 ) ) ).

% DiffD1
thf(fact_3620_DiffD1,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_3621_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_3622_DiffD1,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_3623_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_3624_DiffD2,axiom,
    ! [C: complex,A2: set_complex,B3: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
     => ~ ( member_complex @ C @ B3 ) ) ).

% DiffD2
thf(fact_3625_DiffD2,axiom,
    ! [C: real,A2: set_real,B3: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
     => ~ ( member_real @ C @ B3 ) ) ).

% DiffD2
thf(fact_3626_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
     => ~ ( member_set_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_3627_DiffD2,axiom,
    ! [C: int,A2: set_int,B3: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
     => ~ ( member_int @ C @ B3 ) ) ).

% DiffD2
thf(fact_3628_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
     => ~ ( member_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_3629_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X3: real] : ( member_real @ X3 @ A6 )
            @ ^ [X3: real] : ( member_real @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3630_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ A6 )
            @ ^ [X3: product_prod_int_int] : ( member5262025264175285858nt_int @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3631_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X3: complex] : ( member_complex @ X3 @ A6 )
            @ ^ [X3: complex] : ( member_complex @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3632_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A6 )
            @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3633_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X3: int] : ( member_int @ X3 @ A6 )
            @ ^ [X3: int] : ( member_int @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3634_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A6 )
            @ ^ [X3: nat] : ( member_nat @ X3 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_3635_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( collect_real
          @ ^ [X3: real] :
              ( ( member_real @ X3 @ A6 )
              & ~ ( member_real @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3636_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A6: set_Pr958786334691620121nt_int,B6: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X3: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X3 @ A6 )
              & ~ ( member5262025264175285858nt_int @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3637_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( collect_complex
          @ ^ [X3: complex] :
              ( ( member_complex @ X3 @ A6 )
              & ~ ( member_complex @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3638_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A6: set_set_nat,B6: set_set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] :
              ( ( member_set_nat @ X3 @ A6 )
              & ~ ( member_set_nat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3639_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( collect_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ A6 )
              & ~ ( member_int @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3640_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ A6 )
              & ~ ( member_nat @ X3 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3641_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3642_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( 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] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3643_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( 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] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3644_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_3645_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_3646_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3647_unset__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3648_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_3649_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3650_flip__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3651_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_3652_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A5: real,B5: real,C2: real] :
            ( ( P @ A5 @ B5 )
           => ( ( P @ B5 @ C2 )
             => ( ( ord_less_eq_real @ A5 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C2 )
                 => ( P @ A5 @ C2 ) ) ) ) )
       => ( ! [X6: real] :
              ( ( ord_less_eq_real @ A @ X6 )
             => ( ( ord_less_eq_real @ X6 @ B )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A5: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X6 )
                          & ( ord_less_eq_real @ X6 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
                       => ( P @ A5 @ B5 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_3653_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_3654_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_3655_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_3656_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_3657_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_3658_mult__le__cancel__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3659_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3660_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3661_mult__le__cancel__iff2,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3662_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3663_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3664_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_3665_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_3666_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_3667_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3668_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3669_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3670_product__nth,axiom,
    ! [N: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N )
        = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3671_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3672_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3673_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3674_product__nth,axiom,
    ! [N: nat,Xs: list_o,Ys: list_int] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N )
        = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3675_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3676_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3677_obtain__set__succ,axiom,
    ! [X: nat,Z: nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_nat @ X @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ B3 )
         => ( ( A2 = B3 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_3678_obtain__set__pred,axiom,
    ! [Z: nat,X: nat,A2: set_nat] :
      ( ( ord_less_nat @ Z @ X )
     => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
       => ( ( finite_finite_nat @ A2 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_3679_set__vebt__finite,axiom,
    ! [T: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).

% set_vebt_finite
thf(fact_3680_pred__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ X4 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_3681_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
     => ( ( finite_finite_nat @ Xs )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs )
              & ( ord_less_nat @ A @ X4 ) ) ) ) ).

% succ_none_empty
thf(fact_3682_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_3683_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_3684_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_3685_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_3686_finite__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% finite_atLeastAtMost
thf(fact_3687_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_3688_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_3689_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3690_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3691_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3692_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3693_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3694_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3695_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3696_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3697_length__product,axiom,
    ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3698_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3699_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
       => ( R = R3 ) ) ) ).

% unique_remainder
thf(fact_3700_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
       => ( Q2 = Q4 ) ) ) ).

% unique_quotient
thf(fact_3701_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ N4 )
         => ( ord_less_nat @ X6 @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_3702_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N6 )
         => ( ord_less_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_3703_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M3: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N6 )
         => ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_3704_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3705_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3706_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3707_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_3708_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K2: nat] :
            ( ( P @ K2 )
            & ( ord_less_nat @ K2 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_3709_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_3710_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3711_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3712_finite__lists__length__eq,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_o @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3713_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3714_finite__lists__length__eq,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_int @ Xs2 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_3715_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_3716_div__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( divide_divide_int @ K @ L2 )
        = Q2 ) ) ).

% div_int_unique
thf(fact_3717_mod__int__unique,axiom,
    ! [K: int,L2: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( modulo_modulo_int @ K @ L2 )
        = R ) ) ).

% mod_int_unique
thf(fact_3718_ex__min__if__finite,axiom,
    ! [S2: set_real] :
      ( ( finite_finite_real @ S2 )
     => ( ( S2 != bot_bot_set_real )
       => ? [X6: real] :
            ( ( member_real @ X6 @ S2 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S2 )
                  & ( ord_less_real @ Xa @ X6 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3719_ex__min__if__finite,axiom,
    ! [S2: set_rat] :
      ( ( finite_finite_rat @ S2 )
     => ( ( S2 != bot_bot_set_rat )
       => ? [X6: rat] :
            ( ( member_rat @ X6 @ S2 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S2 )
                  & ( ord_less_rat @ Xa @ X6 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3720_ex__min__if__finite,axiom,
    ! [S2: set_num] :
      ( ( finite_finite_num @ S2 )
     => ( ( S2 != bot_bot_set_num )
       => ? [X6: num] :
            ( ( member_num @ X6 @ S2 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S2 )
                  & ( ord_less_num @ Xa @ X6 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3721_ex__min__if__finite,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( S2 != bot_bot_set_nat )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ S2 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S2 )
                  & ( ord_less_nat @ Xa @ X6 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3722_ex__min__if__finite,axiom,
    ! [S2: set_int] :
      ( ( finite_finite_int @ S2 )
     => ( ( S2 != bot_bot_set_int )
       => ? [X6: int] :
            ( ( member_int @ X6 @ S2 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S2 )
                  & ( ord_less_int @ Xa @ X6 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_3723_infinite__growing,axiom,
    ! [X8: set_real] :
      ( ( X8 != bot_bot_set_real )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ X8 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X8 )
                & ( ord_less_real @ X6 @ Xa ) ) )
       => ~ ( finite_finite_real @ X8 ) ) ) ).

% infinite_growing
thf(fact_3724_infinite__growing,axiom,
    ! [X8: set_rat] :
      ( ( X8 != bot_bot_set_rat )
     => ( ! [X6: rat] :
            ( ( member_rat @ X6 @ X8 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X8 )
                & ( ord_less_rat @ X6 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X8 ) ) ) ).

% infinite_growing
thf(fact_3725_infinite__growing,axiom,
    ! [X8: set_num] :
      ( ( X8 != bot_bot_set_num )
     => ( ! [X6: num] :
            ( ( member_num @ X6 @ X8 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X8 )
                & ( ord_less_num @ X6 @ Xa ) ) )
       => ~ ( finite_finite_num @ X8 ) ) ) ).

% infinite_growing
thf(fact_3726_infinite__growing,axiom,
    ! [X8: set_nat] :
      ( ( X8 != bot_bot_set_nat )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ X8 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X8 )
                & ( ord_less_nat @ X6 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X8 ) ) ) ).

% infinite_growing
thf(fact_3727_infinite__growing,axiom,
    ! [X8: set_int] :
      ( ( X8 != bot_bot_set_int )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ X8 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X8 )
                & ( ord_less_int @ X6 @ Xa ) ) )
       => ~ ( finite_finite_int @ X8 ) ) ) ).

% infinite_growing
thf(fact_3728_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3729_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3730_finite__lists__length__le,axiom,
    ! [A2: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3731_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3732_finite__lists__length__le,axiom,
    ! [A2: set_o,N: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3733_finite__lists__length__le,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3734_finite__lists__length__le,axiom,
    ! [A2: set_int,N: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_3735_eucl__rel__int__dividesI,axiom,
    ! [L2: int,K: int,Q2: int] :
      ( ( L2 != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q2 @ L2 ) )
       => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_3736_eucl__rel__int,axiom,
    ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).

% eucl_rel_int
thf(fact_3737_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_3738_eucl__rel__int__iff,axiom,
    ! [K: int,L2: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R ) )
        & ( ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
            & ( ord_less_int @ R @ L2 ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
         => ( ( ( ord_less_int @ L2 @ zero_zero_int )
             => ( ( ord_less_int @ L2 @ R )
                & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_3739_mult__less__iff1,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3740_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3741_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3742_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_3743_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_3744_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_3745_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3746_finite__Collect__subsets,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3747_finite__Collect__subsets,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_3748_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z3: real] :
              ( ( power_power_real @ Z3 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_3749_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_3750_finite__Diff2,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3751_finite__Diff2,axiom,
    ! [B3: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3752_finite__Diff2,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_3753_finite__Diff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% finite_Diff
thf(fact_3754_finite__Diff,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).

% finite_Diff
thf(fact_3755_finite__Diff,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).

% finite_Diff
thf(fact_3756_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( times_times_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3757_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( times_times_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3758_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( times_times_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3759_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( times_times_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3760_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( times_times_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3761_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( times_times_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3762_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( times_times_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3763_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( times_times_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3764_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( times_times_rat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3765_prod_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( times_times_rat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_3766_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3767_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3768_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3769_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( plus_plus_complex @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3770_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3771_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3772_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I5: int] :
              ( ( member_int @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I5: int] :
                ( ( member_int @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3773_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I5: complex] :
              ( ( member_complex @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I5: complex] :
                ( ( member_complex @ I5 @ I6 )
                & ( ( plus_plus_real @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3774_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I5: real] :
              ( ( member_real @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I5: real] :
                ( ( member_real @ I5 @ I6 )
                & ( ( plus_plus_rat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3775_sum_Ofinite__Collect__op,axiom,
    ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( member_nat @ I5 @ I6 )
              & ( ( X @ I5 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( Y @ I5 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( member_nat @ I5 @ I6 )
                & ( ( plus_plus_rat @ ( X @ I5 ) @ ( Y @ I5 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_3776_finite__atLeastAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L2 @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_3777_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_eq_int @ A @ I5 )
            & ( ord_less_eq_int @ I5 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_3778_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_int @ A @ I5 )
            & ( ord_less_eq_int @ I5 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_3779_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I5: int] :
            ( ( ord_less_eq_int @ A @ I5 )
            & ( ord_less_int @ I5 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_3780_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M5 )
     => ? [N3: nat] :
        ! [X4: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X4 @ M5 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3781_finite__maxlen,axiom,
    ! [M5: set_list_o] :
      ( ( finite_finite_list_o @ M5 )
     => ? [N3: nat] :
        ! [X4: list_o] :
          ( ( member_list_o @ X4 @ M5 )
         => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3782_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M5 )
     => ? [N3: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M5 )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3783_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( ( finite3922522038869484883st_int @ M5 )
     => ? [N3: nat] :
        ! [X4: list_int] :
          ( ( member_list_int @ X4 @ M5 )
         => ( ord_less_nat @ ( size_size_list_int @ X4 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_3784_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X6: real] :
            ( ( member_real @ X6 @ A2 )
            & ( ord_less_eq_real @ A @ X6 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3785_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X6: set_nat] :
            ( ( member_set_nat @ X6 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X6 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3786_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X6: set_int] :
            ( ( member_set_int @ X6 @ A2 )
            & ( ord_less_eq_set_int @ A @ X6 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3787_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X6: rat] :
            ( ( member_rat @ X6 @ A2 )
            & ( ord_less_eq_rat @ A @ X6 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3788_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X6: num] :
            ( ( member_num @ X6 @ A2 )
            & ( ord_less_eq_num @ A @ X6 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3789_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ( ord_less_eq_nat @ A @ X6 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3790_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X6: int] :
            ( ( member_int @ X6 @ A2 )
            & ( ord_less_eq_int @ A @ X6 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_3791_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X6: real] :
            ( ( member_real @ X6 @ A2 )
            & ( ord_less_eq_real @ X6 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3792_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X6: set_nat] :
            ( ( member_set_nat @ X6 @ A2 )
            & ( ord_less_eq_set_nat @ X6 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3793_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X6: set_int] :
            ( ( member_set_int @ X6 @ A2 )
            & ( ord_less_eq_set_int @ X6 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3794_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X6: rat] :
            ( ( member_rat @ X6 @ A2 )
            & ( ord_less_eq_rat @ X6 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3795_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X6: num] :
            ( ( member_num @ X6 @ A2 )
            & ( ord_less_eq_num @ X6 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3796_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ( ord_less_eq_nat @ X6 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3797_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X6: int] :
            ( ( member_int @ X6 @ A2 )
            & ( ord_less_eq_int @ X6 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_3798_finite__subset,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_3799_finite__subset,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B3 )
     => ( ( finite3207457112153483333omplex @ B3 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_3800_finite__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( finite_finite_int @ B3 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_3801_infinite__super,axiom,
    ! [S2: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_3802_infinite__super,axiom,
    ! [S2: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S2 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_3803_infinite__super,axiom,
    ! [S2: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S2 @ T3 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_3804_rev__finite__subset,axiom,
    ! [B3: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_3805_rev__finite__subset,axiom,
    ! [B3: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_3806_rev__finite__subset,axiom,
    ! [B3: set_int,A2: set_int] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ A2 @ B3 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_3807_Diff__infinite__finite,axiom,
    ! [T3: set_int,S2: set_int] :
      ( ( finite_finite_int @ T3 )
     => ( ~ ( finite_finite_int @ S2 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3808_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S2: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S2 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3809_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S2: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_3810_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X6: real] :
            ( ( member_real @ X6 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3811_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X6: set_int] :
            ( ( member_set_int @ X6 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3812_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X6: rat] :
            ( ( member_rat @ X6 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3813_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X6: num] :
            ( ( member_num @ X6 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3814_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3815_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X6: int] :
            ( ( member_int @ X6 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X6 )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_3816_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X6: real] :
            ( ( member_real @ X6 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3817_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X6: set_int] :
            ( ( member_set_int @ X6 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3818_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X6: rat] :
            ( ( member_rat @ X6 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3819_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X6: num] :
            ( ( member_num @ X6 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3820_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3821_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X6: int] :
            ( ( member_int @ X6 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X6 @ Xa )
                 => ( X6 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_3822_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3823_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_3824_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_3825_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_3826_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_3827_prod_Oinject,axiom,
    ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
        = ( produc6137756002093451184nteger @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3828_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y22: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
        = ( produc8603105652947943368nteger @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3829_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3830_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y22: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_3831_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3832_old_Oprod_Oinject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3833_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3834_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3835_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_3836_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A5: int,B5: int] :
          ( Y
         != ( product_Pair_int_int @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_3837_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8763457246119570046nteger] :
      ~ ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( Y
         != ( produc6137756002093451184nteger @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_3838_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( Y
         != ( produc8603105652947943368nteger @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_3839_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_3840_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A5 @ B5 ) ) ).

% old.prod.exhaust
thf(fact_3841_surj__pair,axiom,
    ! [P5: product_prod_int_int] :
    ? [X6: int,Y5: int] :
      ( P5
      = ( product_Pair_int_int @ X6 @ Y5 ) ) ).

% surj_pair
thf(fact_3842_surj__pair,axiom,
    ! [P5: produc8763457246119570046nteger] :
    ? [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
      ( P5
      = ( produc6137756002093451184nteger @ X6 @ Y5 ) ) ).

% surj_pair
thf(fact_3843_surj__pair,axiom,
    ! [P5: produc1908205239877642774nteger] :
    ? [X6: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
      ( P5
      = ( produc8603105652947943368nteger @ X6 @ Y5 ) ) ).

% surj_pair
thf(fact_3844_surj__pair,axiom,
    ! [P5: produc2285326912895808259nt_int] :
    ? [X6: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P5
      = ( produc5700946648718959541nt_int @ X6 @ Y5 ) ) ).

% surj_pair
thf(fact_3845_surj__pair,axiom,
    ! [P5: produc7773217078559923341nt_int] :
    ? [X6: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P5
      = ( produc4305682042979456191nt_int @ X6 @ Y5 ) ) ).

% surj_pair
thf(fact_3846_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P5: product_prod_int_int] :
      ( ! [A5: int,B5: int] : ( P @ ( product_Pair_int_int @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_3847_prod__cases,axiom,
    ! [P: produc8763457246119570046nteger > $o,P5: produc8763457246119570046nteger] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_3848_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P5: produc1908205239877642774nteger] :
      ( ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_3849_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P5: produc2285326912895808259nt_int] :
      ( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_3850_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P5: produc7773217078559923341nt_int] :
      ( ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ B5 ) )
     => ( P @ P5 ) ) ).

% prod_cases
thf(fact_3851_Pair__inject,axiom,
    ! [A: int,B: int,A4: int,B4: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3852_Pair__inject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3853_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3854_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3855_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_3856_prod__cases3,axiom,
    ! [Y: produc8763457246119570046nteger] :
      ~ ! [A5: code_integer > option6357759511663192854e_term,B5: code_integer,C2: code_integer] :
          ( Y
         != ( produc6137756002093451184nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3857_prod__cases3,axiom,
    ! [Y: produc1908205239877642774nteger] :
      ~ ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: code_integer,C2: code_integer] :
          ( Y
         != ( produc8603105652947943368nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3858_prod__cases3,axiom,
    ! [Y: produc2285326912895808259nt_int] :
      ~ ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C2: int] :
          ( Y
         != ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3859_prod__cases3,axiom,
    ! [Y: produc7773217078559923341nt_int] :
      ~ ! [A5: int > option6357759511663192854e_term,B5: int,C2: int] :
          ( Y
         != ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) ) ).

% prod_cases3
thf(fact_3860_prod__induct3,axiom,
    ! [P: produc8763457246119570046nteger > $o,X: produc8763457246119570046nteger] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_3861_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X: produc1908205239877642774nteger] :
      ( ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A5 @ ( produc1086072967326762835nteger @ B5 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_3862_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X: produc2285326912895808259nt_int] :
      ( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_3863_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X: produc7773217078559923341nt_int] :
      ( ! [A5: int > option6357759511663192854e_term,B5: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A5 @ ( product_Pair_int_int @ B5 @ C2 ) ) )
     => ( P @ X ) ) ).

% prod_induct3
thf(fact_3864_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X3 ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_3865_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
             => ( P @ M4 @ N3 ) ) )
       => ( P @ M @ N ) ) ) ).

% gcd_nat_induct
thf(fact_3866_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ 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 @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_3867_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_3868_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_3869_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_3870_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_3871_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_3872_order__refl,axiom,
    ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).

% order_refl
thf(fact_3873_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_3874_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_3875_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_3876_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_3877_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_3878_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X )
            = ( ln_ln_real @ Y ) )
          = ( X = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_3879_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_3880_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_3881_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_3882_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_iff
thf(fact_3883_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = zero_zero_real )
        = ( X = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_3884_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_3885_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_3886_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_3887_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_3888_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_iff
thf(fact_3889_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L2: int,R: int] :
      ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R ) ) ).

% concat_bit_assoc
thf(fact_3890_ln__less__self,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_less_self
thf(fact_3891_ln__bound,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).

% ln_bound
thf(fact_3892_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ one_one_real @ X ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_3893_ln__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_3894_ln__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_gt_zero
thf(fact_3895_ln__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).

% ln_ge_zero
thf(fact_3896_order__antisym__conv,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_3897_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_3898_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_3899_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_3900_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_3901_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_3902_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_3903_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_3904_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_3905_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3906_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3907_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3908_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3909_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3910_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3911_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3912_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3913_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3914_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_3915_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3916_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3917_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3918_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3919_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3920_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3921_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3922_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3923_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3924_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_3925_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_3926_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_3927_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_3928_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_3929_order__eq__refl,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( X = Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_3930_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_3931_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_3932_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_3933_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_3934_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3935_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3936_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3937_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3938_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3939_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3940_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3941_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3942_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3943_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_3944_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3945_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3946_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3947_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_eq_int @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3948_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3949_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3950_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3951_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_eq_int @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3952_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3953_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_3954_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3955_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3956_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3957_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3958_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_3959_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3960_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3961_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3962_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3963_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_3964_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_3965_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_3966_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_3967_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_3968_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_3969_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_3970_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_3971_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_3972_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_3973_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_3974_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3975_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3976_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3977_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3978_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_3979_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat,B5: rat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3980_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num,B5: num] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3981_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3982_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int,B5: int] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_3983_order__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_3984_order__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X @ Z ) ) ) ).

% order_trans
thf(fact_3985_order__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X @ Z ) ) ) ).

% order_trans
thf(fact_3986_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_3987_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_3988_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_3989_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_3990_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_3991_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_3992_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_3993_order__antisym,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_3994_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_3995_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_3996_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_3997_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_3998_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_3999_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_4000_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_4001_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_4002_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_4003_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_4004_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_4005_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_4006_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_4007_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_4008_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z2: set_int] : ( Y4 = Z2 ) )
    = ( ^ [X3: set_int,Y3: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y3 )
          & ( ord_less_eq_set_int @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4009_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z2: rat] : ( Y4 = Z2 ) )
    = ( ^ [X3: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y3 )
          & ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4010_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
    = ( ^ [X3: num,Y3: num] :
          ( ( ord_less_eq_num @ X3 @ Y3 )
          & ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4011_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4012_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_4013_le__cases3,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z ) )
       => ( ( ( ord_less_eq_rat @ X @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4014_le__cases3,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z ) )
       => ( ( ( ord_less_eq_num @ X @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X ) )
             => ~ ( ( ord_less_eq_num @ Z @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4015_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4016_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_4017_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_4018_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_4019_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_4020_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_4021_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_4022_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_4023_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_4024_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_4025_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_4026_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4027_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4028_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4029_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4030_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_4031_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4032_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4033_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4034_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4035_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_4036_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_4037_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_4038_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_4039_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_4040_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_4041_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_4042_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_4043_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_4044_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_4045_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_4046_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_4047_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_4048_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_4049_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_4050_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_4051_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4052_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4053_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4054_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4055_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4056_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4057_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4058_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4059_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4060_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_4061_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4062_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4063_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4064_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4065_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4066_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4067_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4068_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4069_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4070_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_4071_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_4072_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_4073_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_4074_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_4075_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_4076_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4077_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4078_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4079_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4080_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4081_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4082_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4083_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4084_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4085_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_4086_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4087_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4088_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4089_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4090_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4091_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4092_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4093_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4094_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4095_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_4096_order__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4097_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4098_order__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4099_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4100_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_4101_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_4102_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_4103_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_4104_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_4105_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_4106_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_4107_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_4108_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_4109_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_4110_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_4111_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_4112_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_4113_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_4114_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_4115_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_4116_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4117_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4118_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4119_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4120_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_4121_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_4122_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_4123_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_4124_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_4125_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_4126_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_4127_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_4128_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_4129_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_4130_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_4131_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_4132_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_4133_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_4134_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_4135_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_4136_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_4137_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_4138_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_4139_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_4140_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_4141_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_4142_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_4143_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_4144_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_4145_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_4146_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B5: real] :
          ( ( ord_less_real @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: real] : ( P @ A5 @ A5 )
       => ( ! [A5: real,B5: real] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_4147_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B5: rat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_4148_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B5: num] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_4149_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_4150_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B5: int] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_4151_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [N2: nat] :
          ( ( P4 @ N2 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ~ ( P4 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_4152_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_4153_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_4154_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_4155_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_4156_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_4157_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_4158_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_4159_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_4160_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_4161_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_4162_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4163_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4164_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4165_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4166_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_4167_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4168_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4169_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4170_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4171_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_4172_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X6: nat] :
          ( ! [Y2: nat] :
              ( ( ord_less_nat @ Y2 @ X6 )
             => ( P @ Y2 ) )
         => ( P @ X6 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_4173_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_4174_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_4175_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_4176_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_4177_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_4178_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_4179_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_4180_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_4181_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_4182_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_4183_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_4184_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_4185_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_4186_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_4187_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_4188_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4189_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4190_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4191_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4192_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_4193_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z5: real] :
          ( ( ord_less_real @ X @ Z5 )
          & ( ord_less_real @ Z5 @ Y ) ) ) ).

% dense
thf(fact_4194_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z5: rat] :
          ( ( ord_less_rat @ X @ Z5 )
          & ( ord_less_rat @ Z5 @ Y ) ) ) ).

% dense
thf(fact_4195_gt__ex,axiom,
    ! [X: real] :
    ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).

% gt_ex
thf(fact_4196_gt__ex,axiom,
    ! [X: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).

% gt_ex
thf(fact_4197_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_4198_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_4199_lt__ex,axiom,
    ! [X: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X ) ).

% lt_ex
thf(fact_4200_lt__ex,axiom,
    ! [X: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X ) ).

% lt_ex
thf(fact_4201_lt__ex,axiom,
    ! [X: int] :
    ? [Y5: int] : ( ord_less_int @ Y5 @ X ) ).

% lt_ex
thf(fact_4202_small__lazy_H_Ocases,axiom,
    ! [X: product_prod_int_int] :
      ~ ! [D3: int,I4: int] :
          ( X
         != ( product_Pair_int_int @ D3 @ I4 ) ) ).

% small_lazy'.cases
thf(fact_4203_exhaustive__int_H_Ocases,axiom,
    ! [X: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D3: int,I4: int] :
          ( X
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I4 ) ) ) ).

% exhaustive_int'.cases
thf(fact_4204_full__exhaustive__int_H_Ocases,axiom,
    ! [X: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D3: int,I4: int] :
          ( X
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D3 @ I4 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_4205_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_4206_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self
thf(fact_4207_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_4208_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ln_ln_real @ X )
          = ( minus_minus_real @ X @ one_one_real ) )
       => ( X = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_4209_ln__div,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_4210_ln__le__minus__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_4211_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_4212_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4213_order__le__imp__less__or__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4214_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4215_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4216_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4217_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_4218_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_4219_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_4220_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_4221_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_4222_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_4223_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 )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4224_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4225_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4226_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4227_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4228_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 )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4229_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4230_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4231_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4232_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_4233_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4234_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4235_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4236_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4237_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4238_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4239_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4240_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4241_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4242_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_4243_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4244_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4245_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4246_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4247_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4248_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4249_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4250_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4251_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4252_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_eq_num @ X6 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_4253_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 )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4254_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4255_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4256_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4257_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_real @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4258_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 )
       => ( ! [X6: real,Y5: real] :
              ( ( ord_less_real @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4259_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 )
       => ( ! [X6: rat,Y5: rat] :
              ( ( ord_less_rat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4260_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 )
       => ( ! [X6: num,Y5: num] :
              ( ( ord_less_num @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4261_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 )
       => ( ! [X6: nat,Y5: nat] :
              ( ( ord_less_nat @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4262_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 )
       => ( ! [X6: int,Y5: int] :
              ( ( ord_less_int @ X6 @ Y5 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_4263_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4264_order__less__le__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4265_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4266_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4267_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4268_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_4269_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4270_order__le__less__trans,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4271_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4272_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4273_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4274_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_4275_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_4276_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_4277_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_4278_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_4279_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_4280_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_4281_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_4282_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_4283_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_4284_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_4285_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_4286_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_4287_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4288_order__less__imp__le,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4289_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4290_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4291_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4292_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_4293_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4294_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4295_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4296_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4297_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_4298_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4299_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4300_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4301_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4302_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_4303_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y3: real] :
          ( ( ord_less_eq_real @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4304_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X3: set_int,Y3: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4305_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4306_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y3: num] :
          ( ( ord_less_eq_num @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4307_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4308_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_4309_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X3: real,Y3: real] :
          ( ( ord_less_real @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4310_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X3: set_int,Y3: set_int] :
          ( ( ord_less_set_int @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4311_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y3: rat] :
          ( ( ord_less_rat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4312_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X3: num,Y3: num] :
          ( ( ord_less_num @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4313_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4314_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_int @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_4315_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_4316_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_4317_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_4318_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_4319_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_4320_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_4321_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_4322_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_4323_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_4324_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_4325_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_4326_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_4327_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4328_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4329_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4330_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4331_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4332_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_4333_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_4334_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_4335_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_4336_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_4337_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_4338_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_4339_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_4340_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_4341_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_4342_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_4343_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_4344_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_4345_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4346_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4347_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4348_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4349_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4350_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_4351_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_real @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4352_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_set_int @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4353_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_rat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4354_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_num @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4355_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_nat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4356_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_int @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_4357_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X @ W2 )
           => ( ( ord_less_real @ W2 @ Y )
             => ( ord_less_eq_real @ W2 @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_4358_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X @ W2 )
           => ( ( ord_less_rat @ W2 @ Y )
             => ( ord_less_eq_rat @ W2 @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_4359_dense__ge__bounded,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( ord_less_real @ Z @ X )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z @ W2 )
           => ( ( ord_less_real @ W2 @ X )
             => ( ord_less_eq_real @ Y @ W2 ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_4360_dense__ge__bounded,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z @ W2 )
           => ( ( ord_less_rat @ W2 @ X )
             => ( ord_less_eq_rat @ Y @ W2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_4361_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4362_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4363_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4364_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4365_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4366_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_4367_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_4368_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_4369_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_4370_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_4371_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_4372_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_4373_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_4374_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_4375_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_4376_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_4377_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_4378_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_4379_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4380_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4381_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4382_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4383_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4384_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_4385_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_real @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4386_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_set_int @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4387_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_rat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4388_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_num @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4389_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4390_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_int @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_4391_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_4392_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_4393_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_4394_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_4395_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_4396_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X3: real,Y3: real] :
          ( ( ord_less_eq_real @ X3 @ Y3 )
          & ~ ( ord_less_eq_real @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4397_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X3: set_int,Y3: set_int] :
          ( ( ord_less_eq_set_int @ X3 @ Y3 )
          & ~ ( ord_less_eq_set_int @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4398_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X3: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X3 @ Y3 )
          & ~ ( ord_less_eq_rat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4399_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X3: num,Y3: num] :
          ( ( ord_less_eq_num @ X3 @ Y3 )
          & ~ ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4400_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4401_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_4402_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X6: real] :
          ( ( ord_less_real @ X6 @ Y )
         => ( ord_less_eq_real @ X6 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_4403_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X6: rat] :
          ( ( ord_less_rat @ X6 @ Y )
         => ( ord_less_eq_rat @ X6 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_4404_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X6: real] :
          ( ( ord_less_real @ Z @ X6 )
         => ( ord_less_eq_real @ Y @ X6 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_4405_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X6: rat] :
          ( ( ord_less_rat @ Z @ X6 )
         => ( ord_less_eq_rat @ Y @ X6 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_4406_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4407_antisym__conv2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ~ ( ord_less_set_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4408_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4409_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4410_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4411_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_4412_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4413_antisym__conv1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X @ Y )
     => ( ( ord_less_eq_set_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4414_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4415_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4416_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4417_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_4418_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4419_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4420_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4421_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4422_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4423_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_4424_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_4425_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_4426_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_4427_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_4428_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_4429_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_4430_leD,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ~ ( ord_less_set_int @ X @ Y ) ) ).

% leD
thf(fact_4431_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_4432_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_4433_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_4434_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_4435_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_4436_bot_Oextremum__uniqueI,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
     => ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_uniqueI
thf(fact_4437_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_4438_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_4439_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_4440_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_4441_bot_Oextremum__unique,axiom,
    ! [A: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
      = ( A = bot_bo4199563552545308370d_enat ) ) ).

% bot.extremum_unique
thf(fact_4442_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_4443_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_4444_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_4445_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_4446_bot_Oextremum,axiom,
    ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).

% bot.extremum
thf(fact_4447_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_4448_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_4449_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_4450_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_4451_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_4452_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_4453_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_4454_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_4455_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_4456_bot_Oextremum__strict,axiom,
    ! [A: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).

% bot.extremum_strict
thf(fact_4457_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_4458_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_4459_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_4460_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P @ A5 @ B5 )
          = ( P @ B5 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ A5 @ B5 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_4461_max__def,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A3: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4462_max__def,axiom,
    ( ord_max_Code_integer
    = ( ^ [A3: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4463_max__def,axiom,
    ( ord_max_set_int
    = ( ^ [A3: set_int,B2: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4464_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A3: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4465_max__def,axiom,
    ( ord_max_num
    = ( ^ [A3: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4466_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A3: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4467_max__def,axiom,
    ( ord_max_int
    = ( ^ [A3: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B2 ) @ B2 @ A3 ) ) ) ).

% max_def
thf(fact_4468_max__absorb1,axiom,
    ! [Y: extended_enat,X: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Y @ X )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4469_max__absorb1,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Y @ X )
     => ( ( ord_max_Code_integer @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4470_max__absorb1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X )
     => ( ( ord_max_set_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4471_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4472_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4473_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4474_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4475_max__absorb2,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ X @ Y )
     => ( ( ord_ma741700101516333627d_enat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4476_max__absorb2,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ Y )
     => ( ( ord_max_Code_integer @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4477_max__absorb2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ( ord_max_set_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4478_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4479_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4480_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4481_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4482_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_4483_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_4484_tanh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( tanh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_4485_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_4486_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_4487_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4488_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4489_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_4490_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4491_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4492_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
          = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_4493_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4494_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4495_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4496_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4497_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4498_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4499_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4500_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4501_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4502_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4503_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_4504_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_4505_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_4506_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_4507_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_4508_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_4509_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_4510_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_4511_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_4512_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_4513_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_4514_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_4515_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_4516_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4517_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4518_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_4519_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4520_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4521_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4522_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4523_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4524_neg__equal__0__iff__equal,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_4525_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_4526_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4527_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4528_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_4529_neg__0__equal__iff__equal,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A ) )
      = ( zero_zero_real = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4530_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A ) )
      = ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4531_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4532_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4533_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4534_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4535_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4536_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4537_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4538_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4539_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_4540_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_4541_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_4542_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_4543_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4544_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4545_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4546_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4547_neg__numeral__eq__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M = N ) ) ).

% neg_numeral_eq_iff
thf(fact_4548_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_4549_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_4550_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_4551_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_4552_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_4553_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_4554_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_4555_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_4556_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_4557_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_4558_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_4559_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_4560_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_4561_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_4562_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_4563_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_4564_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_4565_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_4566_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_4567_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_4568_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_4569_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_4570_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_4571_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_4572_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_4573_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_4574_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_4575_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_4576_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_4577_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4578_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4579_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_4580_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_4581_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_4582_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4583_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4584_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_4585_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_4586_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_4587_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_4588_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_4589_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_4590_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4591_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4592_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4593_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4594_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4595_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_4596_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_4597_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_4598_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_4599_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_4600_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_4601_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_4602_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_4603_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_4604_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_4605_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_4606_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_4607_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_4608_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_4609_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_4610_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_4611_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_4612_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_4613_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_4614_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_4615_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_4616_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_4617_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_4618_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_4619_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_4620_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_4621_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_4622_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_4623_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_4624_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_4625_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_4626_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_4627_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_4628_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_4629_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4630_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4631_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4632_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4633_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_4634_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4635_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4636_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4637_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4638_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_4639_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_4640_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_4641_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_4642_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_4643_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_4644_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_4645_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_4646_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_4647_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_4648_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_4649_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_4650_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_4651_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_4652_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_4653_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_4654_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_4655_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_4656_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_4657_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_4658_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_4659_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_4660_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_4661_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_4662_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_4663_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_4664_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_4665_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_4666_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_4667_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_4668_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_4669_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4670_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_4671_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_4672_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_4673_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_4674_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_4675_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_4676_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_4677_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_4678_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_4679_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_4680_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_4681_tanh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( tanh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_4682_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% tanh_real_le_iff
thf(fact_4683_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_4684_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4685_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_4686_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_4687_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_4688_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4689_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_4690_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_4691_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_4692_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_4693_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_4694_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_4695_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_4696_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_4697_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_4698_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_4699_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_4700_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_4701_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4702_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_4703_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_4704_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_4705_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4706_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_4707_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_4708_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_4709_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_4710_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_4711_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_4712_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_4713_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_4714_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_4715_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4716_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4717_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4718_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_4719_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4720_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4721_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4722_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4723_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_4724_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4725_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4726_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4727_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4728_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_4729_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_4730_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4731_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4732_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4733_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4734_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_4735_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4736_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4737_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4738_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4739_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4740_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_4741_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_4742_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_4743_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_4744_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_4745_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_4746_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_4747_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_4748_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_4749_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_4750_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_4751_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_4752_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_4753_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_4754_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_4755_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_4756_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_4757_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_4758_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_4759_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_4760_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_4761_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_4762_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4763_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_4764_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_4765_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4766_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_4767_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_4768_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_4769_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_4770_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_4771_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_4772_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_4773_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_4774_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4775_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4776_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4777_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_4778_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_4779_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_4780_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_4781_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_4782_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_4783_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_4784_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_4785_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_4786_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_4787_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_4788_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_4789_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_4790_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_4791_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_4792_divide__minus1,axiom,
    ! [X: real] :
      ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X ) ) ).

% divide_minus1
thf(fact_4793_divide__minus1,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% divide_minus1
thf(fact_4794_divide__minus1,axiom,
    ! [X: rat] :
      ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X ) ) ).

% divide_minus1
thf(fact_4795_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_4796_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_4797_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_4798_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_4799_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_4800_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_4801_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_4802_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_4803_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_4804_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_4805_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_4806_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_4807_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_4808_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_4809_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_4810_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_4811_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_4812_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_4813_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_4814_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_4815_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_4816_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_4817_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_4818_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_4819_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_4820_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_4821_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_4822_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_4823_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_4824_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_4825_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_4826_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_4827_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_4828_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_4829_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_4830_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_neg_numeral
thf(fact_4831_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_neg_numeral
thf(fact_4832_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_4833_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_neg_numeral
thf(fact_4834_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_4835_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_4836_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_4837_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_4838_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_4839_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_4840_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_4841_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_4842_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_4843_abs__power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4844_abs__power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4845_abs__power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4846_abs__power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).

% abs_power_minus
thf(fact_4847_real__add__minus__iff,axiom,
    ! [X: real,A: real] :
      ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X = A ) ) ).

% real_add_minus_iff
thf(fact_4848_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_4849_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% tanh_real_pos_iff
thf(fact_4850_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_4851_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% tanh_real_nonneg_iff
thf(fact_4852_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_4853_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_4854_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4855_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4856_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_4857_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_4858_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_4859_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_4860_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_4861_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_4862_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_4863_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_4864_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_4865_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_4866_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_4867_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_4868_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_4869_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_4870_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_4871_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_4872_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4873_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4874_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4875_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4876_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4877_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4878_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4879_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4880_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4881_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4882_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_4883_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_4884_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_4885_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_4886_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_4887_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_4888_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_4889_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_4890_minus__one__mult__self,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_4891_left__minus__one__mult__self,axiom,
    ! [N: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4892_left__minus__one__mult__self,axiom,
    ! [N: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4893_left__minus__one__mult__self,axiom,
    ! [N: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4894_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4895_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4896_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_4897_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_4898_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_4899_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_4900_even__Suc,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% even_Suc
thf(fact_4901_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_Suc_Suc_iff
thf(fact_4902_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_4903_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_4904_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_4905_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_4906_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_4907_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_4908_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_4909_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_4910_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_4911_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_4912_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_4913_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_4914_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_4915_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_4916_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_4917_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_4918_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_4919_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_4920_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_4921_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_4922_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_4923_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_4924_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4925_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4926_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4927_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4928_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4929_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4930_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4931_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4932_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4933_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4934_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4935_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4936_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4937_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4938_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4939_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4940_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4941_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4942_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4943_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4944_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4945_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4946_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4947_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4948_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4949_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4950_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4951_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4952_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4953_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4954_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4955_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4956_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4957_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4958_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4959_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4960_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4961_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4962_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4963_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4964_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4965_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4966_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4967_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4968_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4969_artanh__minus__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
        = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).

% artanh_minus_real
thf(fact_4970_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4971_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4972_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4973_neg__numeral__le__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M ) ) ).

% neg_numeral_le_iff
thf(fact_4974_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4975_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4976_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4977_neg__numeral__less__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M ) ) ).

% neg_numeral_less_iff
thf(fact_4978_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_4979_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_4980_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_4981_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_4982_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_4983_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_4984_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_4985_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_4986_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_4987_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_4988_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_4989_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_4990_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_4991_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_4992_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_4993_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_4994_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_4995_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_4996_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_4997_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_4998_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_4999_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_5000_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_5001_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_5002_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_5003_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_5004_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_5005_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_5006_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_5007_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_5008_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_5009_power__minus__odd,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5010_power__minus__odd,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5011_power__minus__odd,axiom,
    ! [N: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5012_power__minus__odd,axiom,
    ! [N: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5013_power__minus__odd,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).

% power_minus_odd
thf(fact_5014_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5015_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5016_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
        = ( power_power_complex @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5017_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5018_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_5019_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_5020_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_5021_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_5022_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_5023_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_5024_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5025_zero__less__power__abs__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
      = ( ( A != zero_zero_real )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5026_zero__less__power__abs__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
      = ( ( A != zero_zero_rat )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5027_zero__less__power__abs__iff,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
      = ( ( A != zero_zero_int )
        | ( N = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_5028_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_5029_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_5030_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_5031_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_5032_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_5033_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_5034_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_5035_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_5036_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_5037_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_5038_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_5039_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_5040_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_5041_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_5042_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_5043_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_5044_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_5045_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_5046_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_5047_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_5048_power__less__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_5049_power__less__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_5050_power__less__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_5051_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_5052_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_5053_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_5054_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_5055_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_5056_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_5057_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_5058_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_5059_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_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_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_5069_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_5070_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_5071_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_5072_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_5073_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_5074_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_5075_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_5076_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_5077_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_5078_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_5079_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_5080_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_5081_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_5082_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_5083_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_5084_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_5085_neg__one__even__power,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_5086_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_5087_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_5088_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_5089_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_5090_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% odd_Suc_minus_one
thf(fact_5091_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5092_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5093_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5094_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5095_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_5096_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).

% even_diff_nat
thf(fact_5097_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5098_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5099_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5100_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5101_diff__numeral__special_I3_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).

% diff_numeral_special(3)
thf(fact_5102_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_5103_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_5104_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_5105_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_5106_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_5107_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_5108_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_5109_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5110_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5111_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_5112_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_5113_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_5114_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_5115_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_5116_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_5117_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_5118_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_5119_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_5120_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_5121_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_5122_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_5123_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_5124_even__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_5125_even__power,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_5126_even__power,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% even_power
thf(fact_5127_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_5128_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_5129_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_5130_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_5131_power__minus1__even,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_5132_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_5133_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_5134_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_5135_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_5136_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_5137_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_5138_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_5139_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5140_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5141_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5142_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_5143_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_5144_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_5145_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_5146_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_5147_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_5148_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_5149_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_5150_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_5151_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_5152_dvd__if__abs__eq,axiom,
    ! [L2: real,K: real] :
      ( ( ( abs_abs_real @ L2 )
        = ( abs_abs_real @ K ) )
     => ( dvd_dvd_real @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5153_dvd__if__abs__eq,axiom,
    ! [L2: int,K: int] :
      ( ( ( abs_abs_int @ L2 )
        = ( abs_abs_int @ K ) )
     => ( dvd_dvd_int @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5154_dvd__if__abs__eq,axiom,
    ! [L2: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L2 )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5155_dvd__if__abs__eq,axiom,
    ! [L2: rat,K: rat] :
      ( ( ( abs_abs_rat @ L2 )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L2 @ K ) ) ).

% dvd_if_abs_eq
thf(fact_5156_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5157_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5158_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5159_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_5160_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_5161_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_5162_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_5163_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_5164_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_5165_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_5166_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5167_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5168_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5169_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5170_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5171_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5172_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5173_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_5174_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5175_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5176_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_5177_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_5178_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_5179_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_5180_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_5181_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_5182_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_5183_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_5184_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_5185_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_5186_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_5187_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_5188_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_5189_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_5190_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_5191_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_5192_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_5193_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_5194_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_5195_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_5196_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_5197_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_5198_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_5199_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_5200_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_5201_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_5202_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_5203_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_5204_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_5205_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_5206_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_5207_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_5208_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_5209_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_5210_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_5211_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_5212_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_5213_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_5214_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_5215_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_5216_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_5217_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_5218_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_5219_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_5220_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_real_def
thf(fact_5221_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_5222_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_5223_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_5224_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_5225_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_5226_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_5227_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_5228_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_5229_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_5230_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_5231_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_5232_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_5233_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_5234_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_5235_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_5236_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_5237_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_5238_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_5239_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_5240_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_5241_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_5242_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5243_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5244_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5245_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_5246_tanh__real__gt__neg1,axiom,
    ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).

% tanh_real_gt_neg1
thf(fact_5247_power__abs,axiom,
    ! [A: code_integer,N: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% power_abs
thf(fact_5248_power__abs,axiom,
    ! [A: rat,N: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% power_abs
thf(fact_5249_power__abs,axiom,
    ! [A: real,N: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% power_abs
thf(fact_5250_power__abs,axiom,
    ! [A: int,N: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% power_abs
thf(fact_5251_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_5252_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_5253_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_5254_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_5255_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_5256_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_5257_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_5258_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B2: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B2 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_5259_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B2: real] :
          ( ( A3 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_5260_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( A3 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_5261_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K3: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).

% dvdE
thf(fact_5262_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K3: real] :
            ( A
           != ( times_times_real @ B @ K3 ) ) ) ).

% dvdE
thf(fact_5263_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K3: rat] :
            ( A
           != ( times_times_rat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_5264_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K3: nat] :
            ( A
           != ( times_times_nat @ B @ K3 ) ) ) ).

% dvdE
thf(fact_5265_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K3: int] :
            ( A
           != ( times_times_int @ B @ K3 ) ) ) ).

% dvdE
thf(fact_5266_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_5267_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_5268_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_5269_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_5270_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_5271_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A3: code_integer] :
        ? [K2: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_5272_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A3: real] :
        ? [K2: real] :
          ( A3
          = ( times_times_real @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_5273_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A3: rat] :
        ? [K2: rat] :
          ( A3
          = ( times_times_rat @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_5274_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A3: nat] :
        ? [K2: nat] :
          ( A3
          = ( times_times_nat @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_5275_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A3: int] :
        ? [K2: int] :
          ( A3
          = ( times_times_int @ B2 @ K2 ) ) ) ) ).

% dvd_def
thf(fact_5276_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_5277_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_5278_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_5279_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_5280_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_5281_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_5282_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_5283_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_5284_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_5285_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_5286_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_5287_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_5288_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_5289_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_5290_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_5291_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5292_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5293_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5294_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5295_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5296_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_5297_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_5298_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5299_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_5300_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_5301_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_5302_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_5303_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_5304_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_5305_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_5306_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5307_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5308_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5309_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5310_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5311_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_5312_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_5313_dvd__productE,axiom,
    ! [P5: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P5 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X6: nat,Y5: nat] :
            ( ( P5
              = ( times_times_nat @ X6 @ Y5 ) )
           => ( ( dvd_dvd_nat @ X6 @ A )
             => ~ ( dvd_dvd_nat @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5314_dvd__productE,axiom,
    ! [P5: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P5 @ ( times_times_int @ A @ B ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( times_times_int @ X6 @ Y5 ) )
           => ( ( dvd_dvd_int @ X6 @ A )
             => ~ ( dvd_dvd_int @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5315_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_5316_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_5317_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_5318_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_5319_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_5320_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_5321_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_5322_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_5323_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_5324_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_5325_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_5326_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_5327_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_5328_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_5329_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_5330_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_5331_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_5332_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_5333_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_5334_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_5335_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_5336_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_5337_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_5338_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_5339_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_5340_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_5341_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_5342_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_5343_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_5344_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_5345_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_5346_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_5347_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_5348_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_5349_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_5350_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_5351_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_5352_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_5353_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_5354_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_5355_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_5356_dvd__diff,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( dvd_dvd_Code_integer @ X @ Z )
       => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5357_dvd__diff,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( dvd_dvd_real @ X @ Z )
       => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5358_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5359_dvd__diff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5360_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_5361_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_5362_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_5363_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_5364_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_5365_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_5366_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_5367_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_5368_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_5369_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_5370_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_5371_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_5372_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_5373_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_5374_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_5375_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_5376_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_5377_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_5378_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_5379_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_5380_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_5381_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_5382_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_5383_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_5384_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_5385_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_5386_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_5387_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5388_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5389_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5390_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5391_neg__numeral__neq__numeral,axiom,
    ! [M: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5392_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5393_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5394_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5395_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5396_numeral__neq__neg__numeral,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5397_dvd__power__same,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5398_dvd__power__same,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5399_dvd__power__same,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5400_dvd__power__same,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5401_dvd__power__same,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5402_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_5403_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_5404_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_5405_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_5406_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_5407_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_5408_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_5409_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_5410_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_5411_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_5412_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_5413_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_5414_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_5415_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_5416_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_5417_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_5418_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_5419_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_5420_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_5421_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_5422_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_5423_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_5424_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_5425_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_5426_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_5427_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5428_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5429_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5430_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5431_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5432_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_5433_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_5434_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_5435_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_5436_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_5437_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5438_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5439_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5440_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5441_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5442_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_5443_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_5444_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_5445_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_5446_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5447_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_5448_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_5449_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5450_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_5451_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_5452_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5453_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_5454_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_5455_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_5456_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_5457_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_5458_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_5459_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_5460_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_5461_dvd__mod,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_5462_dvd__mod,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_5463_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).

% dvd_mod
thf(fact_5464_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5465_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5466_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5467_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5468_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5469_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5470_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_5471_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5472_mod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A4 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5473_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A4 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5474_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_5475_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_5476_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_5477_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
     => ( ( dvd_dvd_int @ K @ N )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_5478_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_5479_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_5480_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_5481_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5482_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5483_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5484_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5485_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5486_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5487_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5488_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5489_power__even__abs,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
        = ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5490_power__even__abs,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
        = ( power_power_rat @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5491_power__even__abs,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
        = ( power_power_real @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5492_power__even__abs,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
        = ( power_power_int @ A @ N ) ) ) ).

% power_even_abs
thf(fact_5493_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_5494_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_5495_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_5496_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_5497_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5498_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5499_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5500_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_5501_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_5502_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_5503_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_5504_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_5505_uminus__power__if,axiom,
    ! [N: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( power_power_real @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5506_uminus__power__if,axiom,
    ! [N: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( power_power_int @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5507_uminus__power__if,axiom,
    ! [N: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( power_power_complex @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5508_uminus__power__if,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( power_8256067586552552935nteger @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5509_uminus__power__if,axiom,
    ! [N: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( power_power_rat @ A @ N ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% uminus_power_if
thf(fact_5510_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_5511_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_5512_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_5513_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_5514_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_5515_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_5516_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_5517_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_5518_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_5519_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_5520_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_5521_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_5522_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_5523_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_5524_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_5525_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_5526_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_5527_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_5528_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_5529_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_5530_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_5531_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_5532_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_5533_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_5534_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_5535_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z5 @ X4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5536_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5537_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5538_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5539_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5540_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z5 @ X4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5541_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z5 @ X4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5542_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z5 @ X4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5543_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5544_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z5 @ X4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5545_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z5 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5546_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5547_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5548_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5549_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5550_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z5: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z5 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5551_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z5: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z5 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5552_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z5: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z5 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5553_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5554_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z5: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z5 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5555_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_5556_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_5557_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_5558_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_5559_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_5560_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_5561_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_5562_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_5563_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_5564_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_5565_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_5566_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_5567_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_5568_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_5569_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_5570_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_5571_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_5572_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_5573_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_5574_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_5575_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_5576_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_5577_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_5578_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_5579_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_5580_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_5581_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_5582_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_5583_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_5584_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_5585_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_5586_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_5587_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_5588_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_5589_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_5590_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_5591_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_5592_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_5593_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_5594_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_5595_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_5596_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_5597_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_5598_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_5599_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_5600_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5601_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5602_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5603_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5604_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5605_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5606_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5607_neg__numeral__le__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5608_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5609_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5610_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5611_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5612_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5613_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_5614_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_5615_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_5616_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_5617_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_5618_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_5619_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_5620_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_5621_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_5622_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_5623_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_5624_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_5625_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_5626_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_5627_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_5628_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5629_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5630_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5631_neg__numeral__less__numeral,axiom,
    ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5632_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5633_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5634_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5635_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5636_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_5637_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_5638_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_5639_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_5640_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_5641_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_5642_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_5643_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_5644_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_5645_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_5646_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_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_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_5654_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_5655_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_5656_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5657_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5658_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_5659_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_5660_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_5661_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_5662_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_5663_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_5664_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_5665_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_5666_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_5667_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_5668_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_5669_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5670_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5671_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5672_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5673_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5674_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_5675_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_5676_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_5677_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_5678_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_5679_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_5680_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_5681_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_5682_numeral__times__minus__swap,axiom,
    ! [W: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5683_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5684_numeral__times__minus__swap,axiom,
    ! [W: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5685_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5686_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5687_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% div_power
thf(fact_5688_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% div_power
thf(fact_5689_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% div_power
thf(fact_5690_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_5691_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_5692_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_5693_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_5694_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_5695_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_5696_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5697_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5698_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5699_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5700_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5701_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5702_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5703_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5704_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5705_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5706_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5707_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5708_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5709_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5710_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5711_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5712_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5713_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5714_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5715_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5716_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5717_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5718_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5719_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5720_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5721_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5722_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5723_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5724_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5725_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5726_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5727_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5728_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5729_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5730_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5731_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5732_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5733_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5734_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5735_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5736_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5737_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5738_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5739_le__imp__power__dvd,axiom,
    ! [M: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5740_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5741_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5742_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5743_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5744_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5745_dvd__power__le,axiom,
    ! [X: code_integer,Y: code_integer,N: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5746_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5747_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5748_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5749_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5750_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_5751_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_5752_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5753_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5754_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5755_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_5756_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M @ N )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_5757_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_5758_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
      = ( ( ord_less_nat @ N @ M )
        | ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_minus_self
thf(fact_5759_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N )
       => ( ( dvd_dvd_int @ M @ N )
         => ( ( dvd_dvd_int @ N @ M )
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_5760_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( dvd_dvd_nat @ M @ N )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_5761_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_5762_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
     => ( ( dvd_dvd_nat @ K @ N )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_5763_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N )
       => ~ ( dvd_dvd_int @ N @ M ) ) ) ).

% zdvd_not_zless
thf(fact_5764_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X6: nat,Y5: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X6 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X6 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_5765_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X6: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X6 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X6 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_5766_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_5767_zdvd__mono,axiom,
    ! [K: int,M: int,T: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).

% zdvd_mono
thf(fact_5768_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X6: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X6 ) @ ( times_times_nat @ B @ Y5 ) )
          = D3 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X6 ) @ ( times_times_nat @ A @ Y5 ) )
          = D3 ) ) ) ).

% bezout1_nat
thf(fact_5769_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_5770_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_5771_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N ) ) ).

% zdvd_reduce
thf(fact_5772_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5773_power__mono__even,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5774_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5775_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_5776_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X3: real,Y3: real] : ( plus_plus_real @ X3 @ ( uminus_uminus_real @ Y3 ) ) ) ) ).

% minus_real_def
thf(fact_5777_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_5778_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_5779_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_5780_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_5781_minus__one__power__iff,axiom,
    ! [N: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_5782_tanh__real__lt__1,axiom,
    ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_5783_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_5784_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_5785_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_5786_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_5787_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_5788_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_5789_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5790_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5791_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5792_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_5793_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5794_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_5795_zero__le__power__abs,axiom,
    ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5796_zero__le__power__abs,axiom,
    ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5797_zero__le__power__abs,axiom,
    ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5798_zero__le__power__abs,axiom,
    ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).

% zero_le_power_abs
thf(fact_5799_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_5800_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_5801_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_5802_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_5803_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_5804_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_5805_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_5806_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_5807_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5808_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5809_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5810_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_5811_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5812_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5813_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5814_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_5815_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_5816_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_5817_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L2: code_integer] :
      ( ( ? [X3: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X3 ) ) )
      = ( ? [X3: code_integer] :
            ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X3 @ zero_z3403309356797280102nteger ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5818_unity__coeff__ex,axiom,
    ! [P: complex > $o,L2: complex] :
      ( ( ? [X3: complex] : ( P @ ( times_times_complex @ L2 @ X3 ) ) )
      = ( ? [X3: complex] :
            ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X3 @ zero_zero_complex ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5819_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X3: real] : ( P @ ( times_times_real @ L2 @ X3 ) ) )
      = ( ? [X3: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X3 @ zero_zero_real ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5820_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X3: rat] : ( P @ ( times_times_rat @ L2 @ X3 ) ) )
      = ( ? [X3: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X3 @ zero_zero_rat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5821_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X3: nat] : ( P @ ( times_times_nat @ L2 @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X3 @ zero_zero_nat ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5822_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X3: int] : ( P @ ( times_times_int @ L2 @ X3 ) ) )
      = ( ? [X3: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X3 @ zero_zero_int ) )
            & ( P @ X3 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5823_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5824_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5825_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5826_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_5827_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_5828_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_5829_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_5830_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_5831_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_5832_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_5833_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_5834_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_5835_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_5836_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_5837_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_5838_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_5839_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_5840_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_5841_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5842_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5843_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5844_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_5845_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_5846_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_5847_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_5848_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_5849_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_5850_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_5851_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_5852_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5853_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5854_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5855_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5856_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_5857_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5858_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_5859_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_5860_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_5861_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_5862_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_5863_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_5864_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_5865_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_5866_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_5867_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_5868_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_5869_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_5870_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_5871_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_5872_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_5873_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_5874_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_5875_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_5876_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_5877_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_5878_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_5879_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_5880_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5881_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_5882_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5883_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5884_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5885_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5886_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_5887_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5888_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_5889_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_5890_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_5891_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5892_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_5893_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_5894_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_5895_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_5896_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5897_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_5898_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_5899_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_5900_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5901_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_5902_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_5903_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5904_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_5905_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_5906_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_5907_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5908_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_5909_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_5910_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_5911_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_5912_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_5913_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_5914_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_5915_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5916_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_5917_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_5918_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_5919_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5920_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_5921_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_5922_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_5923_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_5924_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5925_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_5926_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_5927_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_5928_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5929_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_5930_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_5931_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_5932_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5933_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_5934_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_5935_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_5936_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5937_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_5938_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_5939_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_5940_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_5941_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_5942_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5943_is__unit__power__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5944_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5945_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_5946_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_5947_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_5948_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_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_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_5965_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_5966_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_5967_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_5968_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_5969_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_5970_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_5971_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_5972_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_5973_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5974_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5975_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5976_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5977_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5978_power__minus,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).

% power_minus
thf(fact_5979_power__minus,axiom,
    ! [A: int,N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).

% power_minus
thf(fact_5980_power__minus,axiom,
    ! [A: complex,N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_minus
thf(fact_5981_power__minus,axiom,
    ! [A: code_integer,N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% power_minus
thf(fact_5982_power__minus,axiom,
    ! [A: rat,N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_minus
thf(fact_5983_power__minus__Bit0,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5984_power__minus__Bit0,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5985_power__minus__Bit0,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5986_power__minus__Bit0,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5987_power__minus__Bit0,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5988_lemma__interval__lt,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y2: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y2 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y2 )
                  & ( ord_less_real @ Y2 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_5989_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( ( dvd_dvd_nat @ K @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K @ N ) ) ) ).

% dvd_imp_le
thf(fact_5990_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_5991_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
        = ( dvd_dvd_nat @ M @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_5992_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X6: nat,Y5: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X6 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_5993_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( ( dvd_dvd_int @ Z @ N )
     => ( ( ord_less_int @ zero_zero_int @ N )
       => ( ord_less_eq_int @ Z @ N ) ) ) ).

% zdvd_imp_le
thf(fact_5994_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_5995_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_5996_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_less_0_iff
thf(fact_5997_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).

% real_add_le_0_iff
thf(fact_5998_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_5999_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_6000_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K3: nat,M4: nat] :
          ( X
         != ( product_Pair_nat_nat @ K3 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_6001_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_6002_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6003_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6004_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6005_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6006_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_6007_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_6008_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_6009_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6010_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6011_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A )
                   => ( ( ( times_times_int @ A @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_6012_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_6013_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_6014_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_6015_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_6016_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_6017_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_6018_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_6019_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_6020_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_6021_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_6022_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_6023_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_6024_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_6025_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_6026_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_6027_bit__eq__rec,axiom,
    ( ( ^ [Y4: code_integer,Z2: code_integer] : ( Y4 = Z2 ) )
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6028_bit__eq__rec,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6029_bit__eq__rec,axiom,
    ( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
    = ( ^ [A3: int,B2: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_6030_dvd__power__iff,axiom,
    ! [X: code_integer,M: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_6031_dvd__power__iff,axiom,
    ! [X: nat,M: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_6032_dvd__power__iff,axiom,
    ! [X: int,M: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_6033_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_6034_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_6035_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_6036_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_6037_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_6038_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_6039_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_6040_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_6041_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_6042_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_6043_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_6044_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_6045_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_6046_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_6047_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_6048_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_6049_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_6050_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_6051_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6052_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6053_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6054_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_6055_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_6056_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_6057_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_6058_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_rat ) )
     => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).

% dvd_power
thf(fact_6059_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_6060_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).

% dvd_power
thf(fact_6061_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_6062_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).

% dvd_power
thf(fact_6063_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_6064_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_6065_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_6066_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_6067_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_6068_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_6069_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6070_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6071_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6072_power2__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6073_power2__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6074_power2__eq__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6075_power2__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6076_power2__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6077_lemma__interval,axiom,
    ! [A: real,X: real,B: real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y2: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y2 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y2 )
                  & ( ord_less_eq_real @ Y2 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_6078_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_6079_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_6080_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_6081_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_6082_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_6083_mod__nat__eqI,axiom,
    ! [R: nat,N: nat,M: nat] :
      ( ( ord_less_nat @ R @ N )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_6084_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).

% ln_add_one_self_le_self2
thf(fact_6085_mod__int__pos__iff,axiom,
    ! [K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( ( L2 = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).

% mod_int_pos_iff
thf(fact_6086_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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ 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_6087_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 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ 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_6088_bset_I10_J,axiom,
    ! [D: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ 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_6089_bset_I9_J,axiom,
    ! [D: int,D4: int,B3: set_int,T: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ 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_6090_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6091_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6092_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6093_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6094_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_6095_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_6096_abs__square__eq__1,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6097_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_6098_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_6099_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_6100_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_6101_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_6102_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_6103_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_6104_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_6105_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_6106_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_6107_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_6108_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_6109_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_6110_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_6111_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_6112_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_6113_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_6114_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_6115_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_6116_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_6117_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_6118_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_6119_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_6120_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_6121_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_6122_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_6123_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_6124_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_6125_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_6126_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_6127_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_6128_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_6129_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_6130_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_6131_odd__pos,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% odd_pos
thf(fact_6132_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
        = ( ord_less_eq_nat @ M @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_6133_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6134_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6135_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6136_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6137_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6138_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_6139_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_6140_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_6141_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_6142_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_6143_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_6144_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_6145_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_6146_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_6147_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_6148_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_6149_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_6150_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X6: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X6 )
         => ( P @ X6 @ ( power_8256067586552552935nteger @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6151_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X: real] :
      ( ! [X6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X6 )
         => ( P @ X6 @ ( power_power_real @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6152_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X: rat] :
      ( ! [X6: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X6 )
         => ( P @ X6 @ ( power_power_rat @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6153_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X: int] :
      ( ! [X6: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X6 )
         => ( P @ X6 @ ( power_power_int @ X6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6154_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6155_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6156_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6157_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6158_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6159_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6160_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6161_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6162_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6163_abs__square__less__1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6164_abs__square__less__1,axiom,
    ! [X: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6165_abs__square__less__1,axiom,
    ! [X: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6166_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_6167_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_6168_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_6169_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_6170_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_6171_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_6172_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_6173_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_6174_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_6175_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_6176_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_6177_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_6178_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_6179_zero__le__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6180_zero__le__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6181_zero__le__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_6182_zero__le__odd__power,axiom,
    ! [N: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6183_zero__le__odd__power,axiom,
    ! [N: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6184_zero__le__odd__power,axiom,
    ! [N: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_odd_power
thf(fact_6185_zero__le__even__power,axiom,
    ! [N: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_6186_zero__le__even__power,axiom,
    ! [N: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_6187_zero__le__even__power,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).

% zero_le_even_power
thf(fact_6188_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6189_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_6190_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_6191_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_6192_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_6193_minus__power__mult__self,axiom,
    ! [A: real,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6194_minus__power__mult__self,axiom,
    ! [A: int,N: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6195_minus__power__mult__self,axiom,
    ! [A: complex,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6196_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6197_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% minus_power_mult_self
thf(fact_6198_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_6199_zero__less__power__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6200_zero__less__power__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6201_zero__less__power__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
      = ( ( N = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_6202_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_6203_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_6204_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_6205_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_6206_power__minus1__odd,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_6207_eq__diff__eq_H,axiom,
    ! [X: real,Y: real,Z: real] :
      ( ( X
        = ( minus_minus_real @ Y @ Z ) )
      = ( Y
        = ( plus_plus_real @ X @ Z ) ) ) ).

% eq_diff_eq'
thf(fact_6208_even__mask__div__iff_H,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_6209_even__mask__div__iff_H,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_6210_even__mask__div__iff_H,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% even_mask_div_iff'
thf(fact_6211_power__le__zero__eq,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6212_power__le__zero__eq,axiom,
    ! [A: rat,N: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6213_power__le__zero__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_6214_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_6215_even__mask__div__iff,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_6216_even__mask__div__iff,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_6217_even__mask__div__iff,axiom,
    ! [M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N ) ) ) ).

% even_mask_div_iff
thf(fact_6218_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6219_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6220_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6221_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: nat,N: 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 ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_6222_compl__le__compl__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_6223_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6224_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_6225_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_6226_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_6227_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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_6228_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_6229_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_6230_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_6231_intind,axiom,
    ! [I: nat,N: nat,P: nat > $o,X: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_6232_intind,axiom,
    ! [I: nat,N: nat,P: int > $o,X: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_6233_intind,axiom,
    ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( P @ X )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).

% intind
thf(fact_6234_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_6235_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
      = ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_6236_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_6237_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_6238_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_6239_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_6240_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_6241_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_6242_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_6243_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_6244_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_6245_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_6246_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6247_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6248_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6249_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6250_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6251_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_6252_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_6253_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_6254_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_6255_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_6256_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_6257_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6258_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6259_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6260_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6261_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6262_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_6263_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_6264_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_6265_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_6266_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_6267_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_6268_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_6269_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_6270_replicate__eq__replicate,axiom,
    ! [M: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M = N )
        & ( ( M != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_6271_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_6272_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_6273_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_6274_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_6275_zdvd1__eq,axiom,
    ! [X: int] :
      ( ( dvd_dvd_int @ X @ one_one_int )
      = ( ( abs_abs_int @ X )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_6276_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_6277_length__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_6278_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_6279_length__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_6280_arctan__eq__zero__iff,axiom,
    ! [X: real] :
      ( ( ( arctan @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_6281_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_6282_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_6283_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_6284_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_6285_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_6286_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_6287_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_6288_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_6289_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_6290_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_6291_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_6292_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_6293_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_6294_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_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_Suc__0__mod__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( zero_n2687167440665602831ol_nat
        @ ( N
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_6302_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_6303_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_6304_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_6305_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_6306_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_6307_Ball__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6308_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6309_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X3 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_6310_Bex__set__replicate,axiom,
    ! [N: nat,A: int,P: int > $o] :
      ( ( ? [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6311_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6312_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X3 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_6313_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6314_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6315_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6316_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6317_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6318_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_6319_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% zero_less_arctan_iff
thf(fact_6320_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_6321_nth__replicate,axiom,
    ! [I: nat,N: nat,X: nat] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_6322_nth__replicate,axiom,
    ! [I: nat,N: nat,X: int] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_6323_nth__replicate,axiom,
    ! [I: nat,N: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
        = X ) ) ).

% nth_replicate
thf(fact_6324_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_6325_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_arctan_iff
thf(fact_6326_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_6327_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_6328_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_6329_odd__of__bool__self,axiom,
    ! [P5: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P5 ) ) )
      = P5 ) ).

% odd_of_bool_self
thf(fact_6330_odd__of__bool__self,axiom,
    ! [P5: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P5 ) ) )
      = P5 ) ).

% odd_of_bool_self
thf(fact_6331_odd__of__bool__self,axiom,
    ! [P5: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P5 ) ) )
      = P5 ) ).

% odd_of_bool_self
thf(fact_6332_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_6333_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_6334_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_6335_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_6336_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_6337_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_6338_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_6339_one__div__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_6340_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_6341_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_6342_bits__1__div__exp,axiom,
    ! [N: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_6343_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_6344_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_6345_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_6346_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ M @ N )
     => ( ( dvd_dvd_nat @ N @ M )
       => ( M = N ) ) ) ).

% dvd_antisym
thf(fact_6347_of__bool__eq__iff,axiom,
    ! [P5: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P5 )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P5 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6348_of__bool__eq__iff,axiom,
    ! [P5: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P5 )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P5 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6349_of__bool__eq__iff,axiom,
    ! [P5: $o,Q2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P5 )
        = ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P5 = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6350_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I5: int] : ( if_int @ ( ord_less_int @ I5 @ zero_zero_int ) @ ( uminus_uminus_int @ I5 ) @ I5 ) ) ) ).

% zabs_def
thf(fact_6351_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_6352_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_6353_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_6354_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_6355_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_6356_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_6357_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_6358_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_6359_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_6360_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_6361_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_6362_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% arctan_le_iff
thf(fact_6363_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_6364_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_6365_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_6366_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_6367_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_6368_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_6369_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_6370_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_6371_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_6372_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_6373_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P2: $o] : ( if_complex @ P2 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_6374_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P2: $o] : ( if_real @ P2 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_6375_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P2: $o] : ( if_rat @ P2 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_6376_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P2: $o] : ( if_nat @ P2 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_6377_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P2: $o] : ( if_int @ P2 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_6378_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P2: $o] : ( if_Code_integer @ P2 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_6379_split__of__bool,axiom,
    ! [P: complex > $o,P5: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
      = ( ( P5
         => ( P @ one_one_complex ) )
        & ( ~ P5
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_6380_split__of__bool,axiom,
    ! [P: real > $o,P5: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
      = ( ( P5
         => ( P @ one_one_real ) )
        & ( ~ P5
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_6381_split__of__bool,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_rat ) )
        & ( ~ P5
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_6382_split__of__bool,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ( P5
         => ( P @ one_one_nat ) )
        & ( ~ P5
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_6383_split__of__bool,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ( P5
         => ( P @ one_one_int ) )
        & ( ~ P5
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_6384_split__of__bool,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ( P5
         => ( P @ one_one_Code_integer ) )
        & ( ~ P5
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_6385_split__of__bool__asm,axiom,
    ! [P: complex > $o,P5: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_complex ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6386_split__of__bool__asm,axiom,
    ! [P: real > $o,P5: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_real ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6387_split__of__bool__asm,axiom,
    ! [P: rat > $o,P5: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_rat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6388_split__of__bool__asm,axiom,
    ! [P: nat > $o,P5: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_nat ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6389_split__of__bool__asm,axiom,
    ! [P: int > $o,P5: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_int ) )
            | ( ~ P5
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6390_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P5: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P5 ) )
      = ( ~ ( ( P5
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P5
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6391_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).

% dvd_imp_le_int
thf(fact_6392_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X6: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X6 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_6393_replicate__length__same,axiom,
    ! [Xs: list_o,X: $o] :
      ( ! [X6: $o] :
          ( ( member_o @ X6 @ ( set_o2 @ Xs ) )
         => ( X6 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_6394_replicate__length__same,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
         => ( X6 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_6395_replicate__length__same,axiom,
    ! [Xs: list_int,X: int] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ ( set_int2 @ Xs ) )
         => ( X6 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
        = Xs ) ) ).

% replicate_length_same
thf(fact_6396_replicate__eqI,axiom,
    ! [Xs: list_complex,N: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs )
        = N )
     => ( ! [Y5: complex] :
            ( ( member_complex @ Y5 @ ( set_complex2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_complex @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6397_replicate__eqI,axiom,
    ! [Xs: list_real,N: nat,X: real] :
      ( ( ( size_size_list_real @ Xs )
        = N )
     => ( ! [Y5: real] :
            ( ( member_real @ Y5 @ ( set_real2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_real @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6398_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = N )
     => ( ! [Y5: set_nat] :
            ( ( member_set_nat @ Y5 @ ( set_set_nat2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_set_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6399_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N )
     => ( ! [Y5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6400_replicate__eqI,axiom,
    ! [Xs: list_o,N: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs )
        = N )
     => ( ! [Y5: $o] :
            ( ( member_o @ Y5 @ ( set_o2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_o @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6401_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N )
     => ( ! [Y5: nat] :
            ( ( member_nat @ Y5 @ ( set_nat2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6402_replicate__eqI,axiom,
    ! [Xs: list_int,N: nat,X: int] :
      ( ( ( size_size_list_int @ Xs )
        = N )
     => ( ! [Y5: int] :
            ( ( member_int @ Y5 @ ( set_int2 @ Xs ) )
           => ( Y5 = X ) )
       => ( Xs
          = ( replicate_int @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_6403_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_6404_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_6405_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_6406_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_6407_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_6408_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( ( times_times_int @ M @ N )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_6409_minus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( minus_minus_int @ zero_zero_int @ L2 )
      = ( uminus_uminus_int @ L2 ) ) ).

% minus_int_code(2)
thf(fact_6410_zmod__zminus1__not__zero,axiom,
    ! [K: int,L2: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L2 )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_6411_zmod__zminus2__not__zero,axiom,
    ! [K: int,L2: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L2 )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_6412_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_6413_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6414_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_6415_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_6416_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_6417_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_6418_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_6419_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_6420_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_6421_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_6422_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_6423_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_6424_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,N: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_6425_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_6426_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_6427_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_6428_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_6429_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_6430_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_6431_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_6432_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_6433_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_6434_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_6435_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_6436_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_6437_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_6438_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_6439_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_6440_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ( ord_less_eq_nat @ M @ I4 )
            & ( ord_less_nat @ I4 @ N ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N ) )
           => ? [I4: nat] :
                ( ( ord_less_eq_nat @ M @ I4 )
                & ( ord_less_eq_nat @ I4 @ N )
                & ( ( F @ I4 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6441_decr__lemma,axiom,
    ! [D: int,X: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_6442_incr__lemma,axiom,
    ! [D: int,Z: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_6443_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_6444_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_6445_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_6446_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_6447_compl__mono,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).

% compl_mono
thf(fact_6448_compl__le__swap1,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X ) )
     => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_6449_compl__le__swap2,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_6450_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_6451_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_6452_exp__mod__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_6453_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N )
         => ( 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 @ N ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6454_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_6455_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K3: int] :
              ( ( P @ K3 )
             => ( ( K3 != zero_zero_int )
               => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K3: int] :
                ( ( P @ K3 )
               => ( ( K3
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_6456_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N )
         => ( 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 @ N ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6457_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6458_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6459_exp__div__exp__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6460_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_6461_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_6462_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_6463_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_6464_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_6465_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_6466_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_6467_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_6468_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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_6469_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X22: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_6470_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_6471_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X22: num] :
      ( ( size_option_num @ X @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_6472_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_6473_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_6474_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_6475_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_6476_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_6477_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_6478_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_6479_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_6480_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_6481_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% subset_decode_imp_le
thf(fact_6482_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_6483_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_6484_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_6485_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_6486_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_6487_crossproduct__eq,axiom,
    ! [W: real,Y: real,X: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6488_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6489_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6490_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_6491_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_6492_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_6493_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_6494_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_6495_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6496_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6497_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_6498_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X3: nat] :
          ( collect_nat
          @ ^ [N2: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_6499_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( 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_6500_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M3: int] :
          ? [N2: int] :
            ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
            & ( member_int @ N2 @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_6501_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K2: int] :
          ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6502_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K2: int] :
          ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6503_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K2: int] :
          ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6504_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6505_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K2: int] :
          ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6506_one__div__minus__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% one_div_minus_numeral
thf(fact_6507_minus__one__div__numeral,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_div_numeral
thf(fact_6508_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N ) )
      = ( M = N ) ) ).

% semiring_norm(90)
thf(fact_6509_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_6510_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_6511_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_6512_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_6513_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(80)
thf(fact_6514_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(73)
thf(fact_6515_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = zero_zero_int )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_6516_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = zero_zero_real )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_6517_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_6518_of__int__eq__0__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = zero_zero_rat )
      = ( Z = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_6519_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_6520_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_6521_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_6522_of__int__0__eq__iff,axiom,
    ! [Z: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z ) )
      = ( Z = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_6523_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_6524_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_6525_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_6526_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_6527_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6528_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6529_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6530_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N ) )
      = ( Z
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6531_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_6532_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_6533_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_6534_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_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_6543_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_6544_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_6545_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_6546_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_6547_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_6548_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_6549_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_6550_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_6551_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_6552_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_6553_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_6554_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_6555_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_add
thf(fact_6556_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_6557_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_17405671764205052669omplex @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_complex @ ( ring_17405671764205052669omplex @ W ) @ ( ring_17405671764205052669omplex @ Z ) ) ) ).

% of_int_diff
thf(fact_6558_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_diff
thf(fact_6559_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_diff
thf(fact_6560_of__int__diff,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_diff
thf(fact_6561_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(9)
thf(fact_6562_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).

% semiring_norm(7)
thf(fact_6563_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6564_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6565_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6566_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6567_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6568_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6569_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6570_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X ) )
      = ( ( power_power_int @ B @ W )
        = X ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6571_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).

% of_int_power
thf(fact_6572_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).

% of_int_power
thf(fact_6573_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).

% of_int_power
thf(fact_6574_of__int__power,axiom,
    ! [Z: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).

% of_int_power
thf(fact_6575_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_6576_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_6577_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(81)
thf(fact_6578_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(72)
thf(fact_6579_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_6580_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_6581_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_6582_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_6583_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_6584_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_6585_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_6586_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_6587_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_6588_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M @ N ) ) ).

% semiring_norm(79)
thf(fact_6589_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% semiring_norm(74)
thf(fact_6590_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_6591_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_6592_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_6593_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_6594_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_6595_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_6596_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_6597_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_6598_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_6599_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_6600_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_6601_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_6602_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6603_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6604_of__int__le__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6605_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_6606_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_6607_of__int__numeral__le__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_6608_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_6609_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_6610_of__int__numeral__less__iff,axiom,
    ! [N: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_6611_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6612_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6613_of__int__less__numeral__iff,axiom,
    ! [Z: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6614_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_6615_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_6616_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_6617_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_6618_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_6619_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_6620_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_6621_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_6622_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_6623_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_6624_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_6625_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_6626_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6627_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6628_dvd__numeral__simp,axiom,
    ! [M: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_6629_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_6630_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_6631_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_6632_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6633_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6634_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6635_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6636_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6637_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6638_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6639_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6640_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6641_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6642_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6643_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6644_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6645_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6646_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6647_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6648_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6649_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6650_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6651_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6652_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_6653_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_6654_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6655_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6656_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6657_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_6658_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_6659_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6660_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6661_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6662_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_6663_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_6664_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6665_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6666_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6667_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6668_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6669_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6670_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6671_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6672_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6673_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6674_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6675_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6676_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6677_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6678_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6679_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6680_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6681_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6682_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6683_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6684_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6685_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6686_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_6687_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6688_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6689_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6690_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6691_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6692_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6693_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6694_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6695_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6696_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6697_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6698_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6699_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6700_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6701_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6702_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6703_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6704_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6705_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6706_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6707_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6708_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6709_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_6710_mult__of__int__commute,axiom,
    ! [X: int,Y: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( ring_17405671764205052669omplex @ X ) ) ) ).

% mult_of_int_commute
thf(fact_6711_mult__of__int__commute,axiom,
    ! [X: int,Y: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).

% mult_of_int_commute
thf(fact_6712_mult__of__int__commute,axiom,
    ! [X: int,Y: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).

% mult_of_int_commute
thf(fact_6713_mult__of__int__commute,axiom,
    ! [X: int,Y: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).

% mult_of_int_commute
thf(fact_6714_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_6715_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_6716_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_6717_xor__num_Ocases,axiom,
    ! [X: product_prod_num_num] :
      ( ( X
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N3: num] :
            ( X
           != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
       => ( ! [N3: num] :
              ( X
             != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
         => ( ! [M4: num] :
                ( X
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N3: num] :
                  ( X
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
             => ( ! [M4: num,N3: num] :
                    ( X
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
               => ( ! [M4: num] :
                      ( X
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N3: num] :
                        ( X
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
                   => ~ ! [M4: num,N3: num] :
                          ( X
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_6718_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_Bit1
thf(fact_6719_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_6720_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_6721_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_6722_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_6723_eval__nat__numeral_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_6724_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6725_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6726_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6727_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6728_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6729_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6730_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6731_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6732_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6733_power__minus__Bit1,axiom,
    ! [X: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6734_power__minus__Bit1,axiom,
    ! [X: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6735_power__minus__Bit1,axiom,
    ! [X: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6736_power__minus__Bit1,axiom,
    ! [X: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6737_power__minus__Bit1,axiom,
    ! [X: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6738_real__of__int__div4,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) ) ).

% real_of_int_div4
thf(fact_6739_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( ( dvd_dvd_int @ D @ N )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_6740_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).

% numeral_code(3)
thf(fact_6741_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_6742_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_6743_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_6744_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_6745_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_6746_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_6747_power__numeral__odd,axiom,
    ! [Z: rat,W: num] :
      ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6748_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_6749_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_6750_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_6751_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_6752_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_6753_of__int__leD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_leD
thf(fact_6754_of__int__leD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).

% of_int_leD
thf(fact_6755_of__int__leD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).

% of_int_leD
thf(fact_6756_of__int__leD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).

% of_int_leD
thf(fact_6757_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_6758_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_6759_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_6760_of__int__lessD,axiom,
    ! [N: int,X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).

% of_int_lessD
thf(fact_6761_of__int__lessD,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X ) ) ) ).

% of_int_lessD
thf(fact_6762_of__int__lessD,axiom,
    ! [N: int,X: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X ) ) ) ).

% of_int_lessD
thf(fact_6763_of__int__lessD,axiom,
    ! [N: int,X: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
     => ( ( N = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X ) ) ) ).

% of_int_lessD
thf(fact_6764_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_6765_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_6766_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_6767_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_6768_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_6769_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6770_numeral__Bit1__div__2,axiom,
    ! [N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N ) ) ).

% numeral_Bit1_div_2
thf(fact_6771_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6772_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6773_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6774_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6775_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6776_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_6777_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_6778_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_6779_power3__eq__cube,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_6780_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_6781_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_6782_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6783_Suc3__eq__add__3,axiom,
    ! [N: nat] :
      ( ( suc @ ( suc @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).

% Suc3_eq_add_3
thf(fact_6784_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_6785_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_6786_real__of__int__div__aux,axiom,
    ! [X: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_6787_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_6788_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6789_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6790_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6791_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_6792_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_6793_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6794_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_6795_real__of__int__div2,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) ) ).

% real_of_int_div2
thf(fact_6796_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_6797_real__of__int__div3,axiom,
    ! [N: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_6798_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_int_def
thf(fact_6799_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6800_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6801_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_def
thf(fact_6802_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_6803_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_6804_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_6805_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N2: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_6806_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N7: nat] :
              ( ( ord_less_nat @ M4 @ N7 )
              & ( member_nat @ N7 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_6807_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M3: nat] :
          ? [N2: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( member_nat @ N2 @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_6808_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( ( modulo_modulo_nat @ N @ ( 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 @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_6809_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6810_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6811_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6812_floor__exists,axiom,
    ! [X: real] :
    ? [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_exists
thf(fact_6813_floor__exists,axiom,
    ! [X: rat] :
    ? [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_exists
thf(fact_6814_floor__exists1,axiom,
    ! [X: real] :
    ? [X6: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X6 ) @ X )
      & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X6 @ one_one_int ) ) )
      & ! [Y2: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y2 ) @ X )
            & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y2 @ one_one_int ) ) ) )
         => ( Y2 = X6 ) ) ) ).

% floor_exists1
thf(fact_6815_floor__exists1,axiom,
    ! [X: rat] :
    ? [X6: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X6 ) @ X )
      & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X6 @ one_one_int ) ) )
      & ! [Y2: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y2 ) @ X )
            & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y2 @ one_one_int ) ) ) )
         => ( Y2 = X6 ) ) ) ).

% floor_exists1
thf(fact_6816_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_6817_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_6818_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_6819_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6820_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6821_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_6822_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_6823_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6824_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6825_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6826_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6827_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_6828_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_6829_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_6830_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_6831_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6832_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K ) )
      = ( N
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_6833_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_6834_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_6835_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_6836_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_6837_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_6838_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_6839_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_6840_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_6841_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_6842_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_6843_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6844_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_6845_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_6846_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_6847_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_6848_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).

% less_numeral_Suc
thf(fact_6849_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6850_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).

% le_numeral_Suc
thf(fact_6851_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_6852_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_6853_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_6854_max__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_6855_max__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_6856_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6857_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6858_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6859_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_6860_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6861_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_6862_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_6863_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6864_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6865_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_6866_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_6867_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_6868_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6869_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6870_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_6871_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_6872_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_6873_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6874_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6875_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X3: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_6876_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X3: real] : ( plus_plus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_6877_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X3: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_6878_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_6879_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6880_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X3: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X3 @ X3 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6881_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X3: real] : ( minus_minus_real @ ( plus_plus_real @ X3 @ X3 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6882_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X3: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X3 @ X3 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6883_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6884_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_6885_ex__le__of__int,axiom,
    ! [X: real] :
    ? [Z5: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).

% ex_le_of_int
thf(fact_6886_ex__le__of__int,axiom,
    ! [X: rat] :
    ? [Z5: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).

% ex_le_of_int
thf(fact_6887_ex__of__int__less,axiom,
    ! [X: real] :
    ? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).

% ex_of_int_less
thf(fact_6888_ex__of__int__less,axiom,
    ! [X: rat] :
    ? [Z5: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z5 ) @ X ) ).

% ex_of_int_less
thf(fact_6889_ex__less__of__int,axiom,
    ! [X: real] :
    ? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).

% ex_less_of_int
thf(fact_6890_ex__less__of__int,axiom,
    ! [X: rat] :
    ? [Z5: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).

% ex_less_of_int
thf(fact_6891_round__unique,axiom,
    ! [X: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_6892_round__unique,axiom,
    ! [X: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X )
          = Y ) ) ) ).

% round_unique
thf(fact_6893_round__unique_H,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X )
        = N ) ) ).

% round_unique'
thf(fact_6894_round__unique_H,axiom,
    ! [X: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X )
        = N ) ) ).

% round_unique'
thf(fact_6895_of__int__round__abs__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_6896_of__int__round__abs__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_6897_of__int__round__gt,axiom,
    ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_gt
thf(fact_6898_of__int__round__gt,axiom,
    ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_gt
thf(fact_6899_of__int__round__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).

% of_int_round_ge
thf(fact_6900_of__int__round__ge,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).

% of_int_round_ge
thf(fact_6901_of__int__round__le,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6902_of__int__round__le,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6903_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_6904_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_6905_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_6906_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_6907_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_6908_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_6909_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_6910_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_6911_round__mono,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_6912_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_6913_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_6914_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_6915_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6916_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6917_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6918_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X3: int] : X3
          @ ( set_or1266510415728281911st_int @ M @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_6919_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6920_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6921_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6922_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_6923_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6924_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6925_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_6926_sum_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A2 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_6927_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_6928_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_6929_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_6930_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_6931_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_6932_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_6933_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6934_case__prod__conv,axiom,
    ! [F: int > int > int,A: int,B: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6935_sum_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6936_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6937_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6938_sum_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6939_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_6940_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6941_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6942_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6943_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_6944_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6945_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_6946_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_6947_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6948_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6949_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6950_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6951_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6952_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6953_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6954_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_6955_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: int] :
              ( ( member_int @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6956_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: complex] :
              ( ( member_complex @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6957_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ F3 )
             => ( ( F @ X3 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6958_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6959_mask__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6960_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_6961_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_6962_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6963_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6964_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6965_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6966_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6967_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6968_sum_Odelta_H,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6969_sum_Odelta_H,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6970_sum_Odelta_H,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6971_sum_Odelta_H,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6972_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6973_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6974_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_complex )
              @ S2 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6975_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6976_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6977_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_real )
              @ S2 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6978_sum_Odelta,axiom,
    ! [S2: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6979_sum_Odelta,axiom,
    ! [S2: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S2 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6980_sum_Odelta,axiom,
    ! [S2: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6981_sum_Odelta,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ zero_zero_rat )
              @ S2 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6982_sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I5: int] : ( abs_abs_int @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_6983_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( abs_abs_real @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_6984_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_6985_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6986_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_6987_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_6988_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_6989_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6990_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6991_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I5: int] : ( abs_abs_int @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_6992_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( abs_abs_real @ ( F @ I5 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_6993_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_6994_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6995_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6996_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6997_sum_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ( G @ X6 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_6998_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ( G @ X6 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_6999_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( ( G @ X6 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_7000_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( ( G @ X6 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_7001_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups5754745047067104278omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7002_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7003_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups3049146728041665814omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7004_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups5808333547571424918x_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7005_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7006_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A2 )
       != zero_zero_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7007_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > rat,A2: set_complex] :
      ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7008_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7009_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7010_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A2: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_7011_of__int__mask__eq,axiom,
    ! [N: nat] :
      ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_int_mask_eq
thf(fact_7012_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_7013_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_7014_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_7015_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X22: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_7016_old_Oprod_Ocase,axiom,
    ! [F: int > int > int,X1: int,X22: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_7017_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_7018_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_7019_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_7020_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_7021_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_7022_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_7023_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_7024_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_7025_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_7026_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_7027_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_7028_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_7029_sum__distrib__left,axiom,
    ! [R: int,F: int > int,A2: set_int] :
      ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7030_sum__distrib__left,axiom,
    ! [R: complex,F: complex > complex,A2: set_complex] :
      ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7031_sum__distrib__left,axiom,
    ! [R: nat,F: nat > nat,A2: set_nat] :
      ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7032_sum__distrib__left,axiom,
    ! [R: real,F: nat > real,A2: set_nat] :
      ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
        @ A2 ) ) ).

% sum_distrib_left
thf(fact_7033_sum__distrib__right,axiom,
    ! [F: int > int,A2: set_int,R: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
      = ( groups4538972089207619220nt_int
        @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7034_sum__distrib__right,axiom,
    ! [F: complex > complex,A2: set_complex,R: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7035_sum__distrib__right,axiom,
    ! [F: nat > nat,A2: set_nat,R: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
      = ( groups3542108847815614940at_nat
        @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7036_sum__distrib__right,axiom,
    ! [F: nat > real,A2: set_nat,R: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_distrib_right
thf(fact_7037_sum__product,axiom,
    ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I5: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I5 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_7038_sum__product,axiom,
    ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I5: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I5 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_7039_sum__product,axiom,
    ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I5 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_7040_sum__product,axiom,
    ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I5 ) @ ( G @ J3 ) )
            @ B3 )
        @ A2 ) ) ).

% sum_product
thf(fact_7041_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( plus_plus_int @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7042_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( plus_plus_complex @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7043_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : ( plus_plus_nat @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7044_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X3: nat] : ( plus_plus_real @ ( G @ X3 ) @ ( H2 @ X3 ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7045_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X3: int] : ( minus_minus_int @ ( F @ X3 ) @ ( G @ X3 ) )
        @ A2 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7046_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X3: complex] : ( minus_minus_complex @ ( F @ X3 ) @ ( G @ X3 ) )
        @ A2 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7047_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X3: nat] : ( minus_minus_real @ ( F @ X3 ) @ ( G @ X3 ) )
        @ A2 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).

% sum_subtractf
thf(fact_7048_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7049_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_7050_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I5: int] : ( modulo_modulo_int @ ( F @ I5 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7051_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( modulo_modulo_nat @ ( F @ I5 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_7052_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 ) )
     => ~ ! [X6: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X6 @ Y5 ) )
           => ~ ( Q @ ( P @ X6 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_7053_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 ) )
     => ~ ! [X6: nat,Y5: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X6 @ Y5 ) )
           => ~ ( Q @ ( P @ X6 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_7054_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 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( Q @ ( P @ X6 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_7055_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( Q @ ( P @ X6 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_7056_case__prodE2,axiom,
    ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
      ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( Z
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( Q @ ( P @ X6 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_7057_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_7058_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_7059_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X3: int,Y3: int] : ( F @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_7060_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X3: int,Y3: int] : ( F @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_7061_case__prod__eta,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int
        @ ^ [X3: int,Y3: int] : ( F @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_7062_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] :
      ( ! [X6: nat,Y5: nat] :
          ( ( F @ X6 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_7063_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X6: nat,Y5: nat] :
          ( ( F @ X6 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_7064_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X6: int,Y5: int] :
          ( ( F @ X6 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X6 @ Y5 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_7065_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X6: int,Y5: int] :
          ( ( F @ X6 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X6 @ Y5 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_7066_cond__case__prod__eta,axiom,
    ! [F: int > int > int,G: product_prod_int_int > int] :
      ( ! [X6: int,Y5: int] :
          ( ( F @ X6 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X6 @ Y5 ) ) )
     => ( ( produc8211389475949308722nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_7067_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_real @ ( F @ X6 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7068_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_real @ ( F @ X6 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7069_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_real @ ( F @ X6 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7070_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X6 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7071_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X6 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7072_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X6 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7073_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X6 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7074_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X6 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7075_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X6 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7076_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X6 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7077_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7078_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7079_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7080_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7081_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7082_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7083_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7084_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7085_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7086_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7087_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_7088_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_7089_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_7090_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_7091_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_7092_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_7093_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_7094_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_7095_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_7096_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_7097_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X3: real] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7098_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X3: nat] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7099_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X3: int] : ( if_complex @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7100_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X3: real] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7101_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X3: int] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7102_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X3: complex] : ( if_real @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7103_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X3: real] :
                ( ( member_real @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X3: real] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7104_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X3: nat] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7105_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X3: int] :
                ( ( member_int @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X3: int] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7106_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
                & ( P @ X3 ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X3: complex] : ( if_rat @ ( P @ X3 ) @ ( G @ X3 ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7107_mask__nonnegative__int,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).

% mask_nonnegative_int
thf(fact_7108_not__mask__negative__int,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_7109_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_7110_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_7111_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7112_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7113_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7114_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7115_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7116_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7117_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7118_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X3: real] :
                ( ( member_real @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7119_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X3: int] :
                ( ( member_int @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7120_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X3: complex] :
                ( ( member_complex @ X3 @ A2 )
               => ( ( F @ X3 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7121_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X6 ) ) )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7122_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X6 ) ) )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7123_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X6 ) ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7124_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X6 ) ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7125_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7126_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7127_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7128_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7129_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7130_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X6 ) ) )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X6 )
                    & ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7131_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7132_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7133_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7134_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7135_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7136_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A2 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7137_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7138_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ A2 )
           => ( ord_less_eq_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7139_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ A2 )
           => ( ord_less_eq_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7140_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ A2 )
           => ( ord_less_eq_int @ ( F @ X6 ) @ ( G @ X6 ) ) )
       => ( ? [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_7141_sum_Orelated,axiom,
    ! [R4: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R4 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7142_sum_Orelated,axiom,
    ! [R4: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
      ( ( R4 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7143_sum_Orelated,axiom,
    ! [R4: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
      ( ( R4 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7144_sum_Orelated,axiom,
    ! [R4: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( R4 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7145_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7146_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7147_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7148_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S2 )
         => ( ! [X6: int] :
                ( ( member_int @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7149_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S2 )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7150_sum_Orelated,axiom,
    ! [R4: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
      ( ( R4 @ zero_zero_int @ zero_zero_int )
     => ( ! [X15: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R4 @ X15 @ X23 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_int @ X15 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S2 )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S2 )
               => ( R4 @ ( H2 @ X6 ) @ ( G @ X6 ) ) )
           => ( R4 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).

% sum.related
thf(fact_7151_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
             => ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7152_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ A2 )
             => ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7153_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ A2 )
             => ( ord_less_real @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7154_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7155_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ A2 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7156_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ A2 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7157_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ A2 )
             => ( ord_less_rat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7158_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ A2 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7159_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ A2 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7160_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ A2 )
             => ( ord_less_nat @ ( F @ X6 ) @ ( G @ X6 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7161_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S2 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7162_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S2 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7163_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S2 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7164_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_complex ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_complex ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S2 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7165_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S2 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7166_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S2 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7167_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A5: real] :
              ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_real @ ( I @ B5 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
               => ( ! [A5: real] :
                      ( ( member_real @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: real] :
                          ( ( member_real @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S2 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7168_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: real] :
                    ( ( member_real @ B5 @ ( minus_minus_set_real @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: real] :
                        ( ( member_real @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S2 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7169_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
           => ( ! [B5: int] :
                  ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: int] :
                    ( ( member_int @ B5 @ ( minus_minus_set_int @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: int] :
                        ( ( member_int @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S2 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7170_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A5: int] :
              ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
             => ( ( I @ ( J @ A5 ) )
                = A5 ) )
         => ( ! [A5: int] :
                ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
               => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                 => ( ( J @ ( I @ B5 ) )
                    = B5 ) )
             => ( ! [B5: complex] :
                    ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
                   => ( member_int @ ( I @ B5 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
               => ( ! [A5: int] :
                      ( ( member_int @ A5 @ S4 )
                     => ( ( G @ A5 )
                        = zero_zero_real ) )
                 => ( ! [B5: complex] :
                        ( ( member_complex @ B5 @ T4 )
                       => ( ( H2 @ B5 )
                          = zero_zero_real ) )
                   => ( ! [A5: int] :
                          ( ( member_int @ A5 @ S2 )
                         => ( ( H2 @ ( J @ A5 ) )
                            = ( G @ A5 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S2 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7171_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B3: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7172_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B3: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7173_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B3: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7174_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B3: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7175_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B3: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B3 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7176_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B3: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7177_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B3: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7178_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B3: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B3 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7179_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > nat,B3: nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = B3 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7180_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > nat,B3: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = B3 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7181_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7182_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7183_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7184_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7185_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I4: nat] :
            ( ( member_nat @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7186_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7187_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7188_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I4: real] :
            ( ( member_real @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7189_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I4: int] :
            ( ( member_int @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7190_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I4: complex] :
            ( ( member_complex @ I4 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7191_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7192_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7193_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7194_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7195_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7196_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X3: int] :
                  ( ( G @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7197_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7198_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X3: complex] :
                  ( ( G @ X3 )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7199_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( G @ X3 )
                  = zero_zero_complex ) ) ) )
        = ( groups2073611262835488442omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7200_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( minus_minus_set_nat @ A2
            @ ( collect_nat
              @ ^ [X3: nat] :
                  ( ( G @ X3 )
                  = zero_zero_rat ) ) ) )
        = ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7201_less__mask,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% less_mask
thf(fact_7202_eval__nat__numeral_I2_J,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_7203_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_7204_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_7205_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_7206_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_7207_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_7208_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_7209_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_7210_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_7211_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_7212_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_7213_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_7214_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_7215_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_7216_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_7217_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_7218_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_7219_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_7220_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_7221_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_7222_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_7223_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_7224_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_7225_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_7226_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7227_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7228_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7229_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7230_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7231_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_nat ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7232_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_nat ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
              = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7233_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ T3 )
              = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7234_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ T3 )
              = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7235_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ T3 )
              = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7236_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_complex ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S2 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7237_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_real ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S2 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7238_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_real ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S2 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7239_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_rat ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S2 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7240_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_rat ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S2 )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7241_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_nat ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S2 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7242_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_nat ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ S2 )
              = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7243_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S2 @ T3 )
       => ( ! [X6: real] :
              ( ( member_real @ X6 @ ( minus_minus_set_real @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_int ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ S2 )
              = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7244_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_int ) )
         => ( ! [X6: complex] :
                ( ( member_complex @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ S2 )
              = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7245_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( H2 @ X6 )
                = zero_zero_complex ) )
         => ( ! [X6: nat] :
                ( ( member_nat @ X6 @ S2 )
               => ( ( G @ X6 )
                  = ( H2 @ X6 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ S2 )
              = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7246_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7247_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7248_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7249_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7250_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7251_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7252_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7253_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T3 )
            = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7254_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T3 )
            = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7255_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T3 )
            = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7256_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S2 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7257_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S2 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7258_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S2 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7259_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S2: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
       => ( ! [X6: complex] :
              ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S2 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7260_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S2 )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7261_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S2 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7262_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S2: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S2 @ T3 )
       => ( ! [X6: nat] :
              ( ( member_nat @ X6 @ ( minus_minus_set_nat @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S2 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7263_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S2 )
            = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7264_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S2 )
            = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7265_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S2: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S2 @ T3 )
       => ( ! [X6: int] :
              ( ( member_int @ X6 @ ( minus_minus_set_int @ T3 @ S2 ) )
             => ( ( G @ X6 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S2 )
            = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7266_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7267_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7268_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7269_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7270_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7271_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7272_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
               => ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7273_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_int ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) )
               => ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7274_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_int ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
               => ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7275_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A2 @ C4 )
       => ( ( ord_less_eq_set_nat @ B3 @ C4 )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: nat] :
                  ( ( member_nat @ B5 @ ( minus_minus_set_nat @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) )
               => ( ( groups2073611262835488442omplex @ G @ A2 )
                  = ( groups2073611262835488442omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7276_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B3 ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7277_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B3 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7278_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_real ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B3 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7279_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7280_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_rat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7281_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7282_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_nat ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B3 ) )
                = ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7283_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B3: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B3 @ C4 )
         => ( ! [A5: real] :
                ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_int ) )
           => ( ! [B5: real] :
                  ( ( member_real @ B5 @ ( minus_minus_set_real @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B3 ) )
                = ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7284_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B3: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B3 @ C4 )
         => ( ! [A5: complex] :
                ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_int ) )
           => ( ! [B5: complex] :
                  ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B3 ) )
                = ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7285_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B3: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A2 @ C4 )
       => ( ( ord_less_eq_set_nat @ B3 @ C4 )
         => ( ! [A5: nat] :
                ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
               => ( ( G @ A5 )
                  = zero_zero_complex ) )
           => ( ! [B5: nat] :
                  ( ( member_nat @ B5 @ ( minus_minus_set_nat @ C4 @ B3 ) )
                 => ( ( H2 @ B5 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ A2 )
                  = ( groups2073611262835488442omplex @ H2 @ B3 ) )
                = ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7286_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7287_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7288_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7289_sum_Osubset__diff,axiom,
    ! [B3: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B3 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7290_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7291_sum_Osubset__diff,axiom,
    ! [B3: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7292_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7293_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7294_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7295_sum_Osubset__diff,axiom,
    ! [B3: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B3 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7296_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7297_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7298_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7299_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7300_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7301_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7302_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7303_sum__diff,axiom,
    ! [A2: set_int,B3: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7304_sum__diff,axiom,
    ! [A2: set_complex,B3: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7305_sum__diff,axiom,
    ! [A2: set_nat,B3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).

% sum_diff
thf(fact_7306_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7307_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B5 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7308_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7309_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7310_sum__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( ! [B5: nat] :
              ( ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B5 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7311_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7312_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B5 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7313_sum__mono2,axiom,
    ! [B3: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ! [B5: real] :
              ( ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7314_sum__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ! [B5: complex] :
              ( ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
         => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7315_sum__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( ! [B5: nat] :
              ( ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B5 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).

% sum_mono2
thf(fact_7316_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_7317_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bitM @ N ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_7318_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_7319_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_7320_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7321_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7322_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_7323_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X6: real] :
                  ( ( member_real @ X6 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7324_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X6: complex] :
                  ( ( member_complex @ X6 @ B3 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7325_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X6: real] :
                  ( ( member_real @ X6 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7326_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X6: complex] :
                  ( ( member_complex @ X6 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7327_sum__strict__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X6: nat] :
                  ( ( member_nat @ X6 @ B3 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7328_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X6: real] :
                  ( ( member_real @ X6 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7329_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X6: complex] :
                  ( ( member_complex @ X6 @ B3 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7330_sum__strict__mono2,axiom,
    ! [B3: set_real,A2: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B3 )
     => ( ( ord_less_eq_set_real @ A2 @ B3 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X6: real] :
                  ( ( member_real @ X6 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X6 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7331_sum__strict__mono2,axiom,
    ! [B3: set_complex,A2: set_complex,B: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X6: complex] :
                  ( ( member_complex @ X6 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X6 ) ) )
             => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7332_sum__strict__mono2,axiom,
    ! [B3: set_nat,A2: set_nat,B: nat,F: nat > int] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A2 @ B3 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X6: nat] :
                  ( ( member_nat @ X6 @ B3 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X6 ) ) )
             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7333_Suc__mask__eq__exp,axiom,
    ! [N: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_mask_eq_exp
thf(fact_7334_mask__nat__less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% mask_nat_less_exp
thf(fact_7335_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ 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
                  @ ^ [I5: complex] : ( times_3573771949741848930nteger @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7336_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ 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
                  @ ^ [I5: real] : ( times_3573771949741848930nteger @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7337_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ 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
                  @ ^ [I5: nat] : ( times_3573771949741848930nteger @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7338_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ 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
                  @ ^ [I5: int] : ( times_3573771949741848930nteger @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7339_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ 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
                  @ ^ [I5: complex] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7340_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ 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
                  @ ^ [I5: real] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7341_convex__sum__bound__le,axiom,
    ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I6 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ 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
                  @ ^ [I5: int] : ( times_times_real @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7342_convex__sum__bound__le,axiom,
    ! [I6: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ I6 )
          = one_one_rat )
       => ( ! [I4: complex] :
              ( ( member_complex @ 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
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I5: complex] : ( times_times_rat @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7343_convex__sum__bound__le,axiom,
    ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ 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
                  @ ^ [I5: real] : ( times_times_rat @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7344_convex__sum__bound__le,axiom,
    ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I6 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ 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
                  @ ^ [I5: nat] : ( times_times_rat @ ( A @ I5 ) @ ( X @ I5 ) )
                  @ I6 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7345_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7346_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7347_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7348_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_7349_mask__half__int,axiom,
    ! [N: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_7350_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_7351_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_7352_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_7353_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_7354_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_7355_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_7356_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_7357_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_7358_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_7359_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_7360_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_7361_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_7362_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( K2 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( uminus_uminus_int @ one_one_int ) )
              @ K2
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_7363_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_7364_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_7365_and_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.left_idem
thf(fact_7366_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.left_idem
thf(fact_7367_and_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.right_idem
thf(fact_7368_and_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.right_idem
thf(fact_7369_case__prodI2,axiom,
    ! [P5: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc6137756002093451184nteger @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc127349428274296955eger_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_7370_case__prodI2,axiom,
    ! [P5: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A5: produc6241069584506657477e_term > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc8603105652947943368nteger @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc6253627499356882019eger_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_7371_case__prodI2,axiom,
    ! [P5: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A5: produc8551481072490612790e_term > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( ( P5
            = ( produc5700946648718959541nt_int @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc1573362020775583542_int_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_7372_case__prodI2,axiom,
    ! [P5: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( ( P5
            = ( produc4305682042979456191nt_int @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc2558449545302689196_int_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_7373_case__prodI2,axiom,
    ! [P5: product_prod_int_int,C: int > int > $o] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( C @ A5 @ B5 ) )
     => ( produc4947309494688390418_int_o @ C @ P5 ) ) ).

% case_prodI2
thf(fact_7374_case__prodI,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_7375_case__prodI,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_7376_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_7377_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_7378_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_7379_mem__case__prodI2,axiom,
    ! [P5: product_prod_int_int,Z: complex,C: int > int > set_complex] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7380_mem__case__prodI2,axiom,
    ! [P5: product_prod_int_int,Z: real,C: int > int > set_real] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( member_real @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7381_mem__case__prodI2,axiom,
    ! [P5: product_prod_int_int,Z: nat,C: int > int > set_nat] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7382_mem__case__prodI2,axiom,
    ! [P5: product_prod_int_int,Z: int,C: int > int > set_int] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( member_int @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7383_mem__case__prodI2,axiom,
    ! [P5: product_prod_int_int,Z: set_nat,C: int > int > set_set_nat] :
      ( ! [A5: int,B5: int] :
          ( ( P5
            = ( product_Pair_int_int @ A5 @ B5 ) )
         => ( member_set_nat @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7384_mem__case__prodI2,axiom,
    ! [P5: produc8763457246119570046nteger,Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc6137756002093451184nteger @ A5 @ B5 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7385_mem__case__prodI2,axiom,
    ! [P5: produc8763457246119570046nteger,Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc6137756002093451184nteger @ A5 @ B5 ) )
         => ( member_real @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7386_mem__case__prodI2,axiom,
    ! [P5: produc8763457246119570046nteger,Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc6137756002093451184nteger @ A5 @ B5 ) )
         => ( member_nat @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7387_mem__case__prodI2,axiom,
    ! [P5: produc8763457246119570046nteger,Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
      ( ! [A5: code_integer > option6357759511663192854e_term,B5: produc8923325533196201883nteger] :
          ( ( P5
            = ( produc6137756002093451184nteger @ A5 @ B5 ) )
         => ( member_int @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7388_mem__case__prodI2,axiom,
    ! [P5: produc7773217078559923341nt_int,Z: complex,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex] :
      ( ! [A5: int > option6357759511663192854e_term,B5: product_prod_int_int] :
          ( ( P5
            = ( produc4305682042979456191nt_int @ A5 @ B5 ) )
         => ( member_complex @ Z @ ( C @ A5 @ B5 ) ) )
     => ( member_complex @ Z @ ( produc7959293469001253456omplex @ C @ P5 ) ) ) ).

% mem_case_prodI2
thf(fact_7389_mem__case__prodI,axiom,
    ! [Z: complex,C: int > int > set_complex,A: int,B: int] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7390_mem__case__prodI,axiom,
    ! [Z: real,C: int > int > set_real,A: int,B: int] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7391_mem__case__prodI,axiom,
    ! [Z: nat,C: int > int > set_nat,A: int,B: int] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7392_mem__case__prodI,axiom,
    ! [Z: int,C: int > int > set_int,A: int,B: int] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7393_mem__case__prodI,axiom,
    ! [Z: set_nat,C: int > int > set_set_nat,A: int,B: int] :
      ( ( member_set_nat @ Z @ ( C @ A @ B ) )
     => ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7394_mem__case__prodI,axiom,
    ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7395_mem__case__prodI,axiom,
    ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_real @ Z @ ( C @ A @ B ) )
     => ( member_real @ Z @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7396_mem__case__prodI,axiom,
    ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_nat @ Z @ ( C @ A @ B ) )
     => ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7397_mem__case__prodI,axiom,
    ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_int @ Z @ ( C @ A @ B ) )
     => ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7398_mem__case__prodI,axiom,
    ! [Z: complex,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_complex @ Z @ ( C @ A @ B ) )
     => ( member_complex @ Z @ ( produc7959293469001253456omplex @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_7399_case__prodI2_H,axiom,
    ! [P5: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ( product_Pair_nat_nat @ A5 @ B5 )
            = P5 )
         => ( C @ A5 @ B5 @ X ) )
     => ( produc8739625826339149834_nat_o @ C @ P5 @ X ) ) ).

% case_prodI2'
thf(fact_7400_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_7401_take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_7402_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_7403_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_7404_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_7405_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_7406_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_7407_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_7408_take__bit__and,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) ) ).

% take_bit_and
thf(fact_7409_take__bit__and,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) ) ).

% take_bit_and
thf(fact_7410_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_7411_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] :
      ( ( bit_concat_bit @ N @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_7412_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_7413_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_7414_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_7415_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_7416_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_7417_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_7418_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_7419_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_7420_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_7421_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_7422_bit_Oconj__one__right,axiom,
    ! [X: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X ) ).

% bit.conj_one_right
thf(fact_7423_bit_Oconj__one__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ ( uminus_uminus_int @ one_one_int ) )
      = X ) ).

% bit.conj_one_right
thf(fact_7424_take__bit__numeral__1,axiom,
    ! [L2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_7425_take__bit__numeral__1,axiom,
    ! [L2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_7426_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ X )
        = one_one_real )
      = ( X = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_7427_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_7428_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_7429_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
        = zero_zero_int )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_7430_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
        = zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
thf(fact_7431_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_7432_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_7433_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_7434_and__numerals_I8_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_7435_one__less__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% one_less_exp_iff
thf(fact_7436_exp__less__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_7437_exp__le__one__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_7438_one__le__exp__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% one_le_exp_iff
thf(fact_7439_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_7440_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_7441_exp__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( exp_real @ ( ln_ln_real @ X ) )
        = X ) ) ).

% exp_ln
thf(fact_7442_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X ) )
        = X )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% exp_ln_iff
thf(fact_7443_take__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_Suc_0
thf(fact_7444_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_7445_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_7446_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_7447_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_7448_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_7449_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_7450_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7451_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7452_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7453_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7454_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7455_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7456_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7457_take__bit__of__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% take_bit_of_1
thf(fact_7458_and__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_7459_and__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_7460_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_7461_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I5: nat] : ( times_times_rat @ ( C @ I5 ) @ ( power_power_rat @ zero_zero_rat @ I5 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I5: nat] : ( times_times_rat @ ( C @ I5 ) @ ( power_power_rat @ zero_zero_rat @ I5 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_7462_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_7463_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7464_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_7465_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7466_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_7467_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7468_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7469_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_7470_and__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_7471_and__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_7472_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
            @ ^ [I5: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I5: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I5 ) @ ( power_power_complex @ zero_zero_complex @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_7473_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
            @ ^ [I5: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I5 ) @ ( power_power_rat @ zero_zero_rat @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I5: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I5 ) @ ( power_power_rat @ zero_zero_rat @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_7474_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
            @ ^ [I5: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I5 ) @ ( power_power_real @ zero_zero_real @ I5 ) ) @ ( D @ I5 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_7475_take__bit__Suc__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_7476_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_7477_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_7478_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7479_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_7480_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7481_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7482_take__bit__of__exp,axiom,
    ! [M: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_7483_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7484_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7485_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_7486_take__bit__eq__mask,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_7487_take__bit__eq__mask,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ) ).

% take_bit_eq_mask
thf(fact_7488_of__int__and__eq,axiom,
    ! [K: int,L2: int] :
      ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
      = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L2 ) ) ) ).

% of_int_and_eq
thf(fact_7489_take__bit__of__int,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_of_int
thf(fact_7490_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.assoc
thf(fact_7491_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.assoc
thf(fact_7492_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_7493_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_7494_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.left_commute
thf(fact_7495_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.left_commute
thf(fact_7496_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_7497_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_7498_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_7499_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = ( bit_se2925701944663578781it_nat @ N @ B ) )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_7500_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_7501_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_7502_exp__not__eq__zero,axiom,
    ! [X: complex] :
      ( ( exp_complex @ X )
     != zero_zero_complex ) ).

% exp_not_eq_zero
thf(fact_7503_exp__not__eq__zero,axiom,
    ! [X: real] :
      ( ( exp_real @ X )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_7504_mem__case__prodE,axiom,
    ! [Z: complex,C: int > int > set_complex,P5: product_prod_int_int] :
      ( ( member_complex @ Z @ ( produc8580519160106071146omplex @ C @ P5 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( member_complex @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7505_mem__case__prodE,axiom,
    ! [Z: real,C: int > int > set_real,P5: product_prod_int_int] :
      ( ( member_real @ Z @ ( produc6452406959799940328t_real @ C @ P5 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( member_real @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7506_mem__case__prodE,axiom,
    ! [Z: nat,C: int > int > set_nat,P5: product_prod_int_int] :
      ( ( member_nat @ Z @ ( produc4251311855443802252et_nat @ C @ P5 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7507_mem__case__prodE,axiom,
    ! [Z: int,C: int > int > set_int,P5: product_prod_int_int] :
      ( ( member_int @ Z @ ( produc73460835934605544et_int @ C @ P5 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7508_mem__case__prodE,axiom,
    ! [Z: set_nat,C: int > int > set_set_nat,P5: product_prod_int_int] :
      ( ( member_set_nat @ Z @ ( produc5233655623923918146et_nat @ C @ P5 ) )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( member_set_nat @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7509_mem__case__prodE,axiom,
    ! [Z: complex,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_complex,P5: produc8763457246119570046nteger] :
      ( ( member_complex @ Z @ ( produc2592262431452330817omplex @ C @ P5 ) )
     => ~ ! [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc6137756002093451184nteger @ X6 @ Y5 ) )
           => ~ ( member_complex @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7510_mem__case__prodE,axiom,
    ! [Z: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P5: produc8763457246119570046nteger] :
      ( ( member_real @ Z @ ( produc815715089573277247t_real @ C @ P5 ) )
     => ~ ! [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc6137756002093451184nteger @ X6 @ Y5 ) )
           => ~ ( member_real @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7511_mem__case__prodE,axiom,
    ! [Z: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P5: produc8763457246119570046nteger] :
      ( ( member_nat @ Z @ ( produc3558942015123893603et_nat @ C @ P5 ) )
     => ~ ! [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc6137756002093451184nteger @ X6 @ Y5 ) )
           => ~ ( member_nat @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7512_mem__case__prodE,axiom,
    ! [Z: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P5: produc8763457246119570046nteger] :
      ( ( member_int @ Z @ ( produc8604463032469472703et_int @ C @ P5 ) )
     => ~ ! [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc6137756002093451184nteger @ X6 @ Y5 ) )
           => ~ ( member_int @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7513_mem__case__prodE,axiom,
    ! [Z: complex,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_complex,P5: produc7773217078559923341nt_int] :
      ( ( member_complex @ Z @ ( produc7959293469001253456omplex @ C @ P5 ) )
     => ~ ! [X6: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P5
              = ( produc4305682042979456191nt_int @ X6 @ Y5 ) )
           => ~ ( member_complex @ Z @ ( C @ X6 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7514_take__bit__minus,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_7515_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_7516_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_7517_take__bit__mult,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_7518_take__bit__diff,axiom,
    ! [N: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_7519_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L2: int,R: int,S: int] :
      ( ( ( bit_concat_bit @ N @ K @ L2 )
        = ( bit_concat_bit @ N @ R @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N @ K )
          = ( bit_se2923211474154528505it_int @ N @ R ) )
        & ( L2 = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_7520_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B: int] :
      ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
      = ( bit_concat_bit @ N @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_7521_case__prodE,axiom,
    ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P5: produc8763457246119570046nteger] :
      ( ( produc127349428274296955eger_o @ C @ P5 )
     => ~ ! [X6: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc6137756002093451184nteger @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7522_case__prodE,axiom,
    ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P5: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C @ P5 )
     => ~ ! [X6: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P5
              = ( produc8603105652947943368nteger @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7523_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P5: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P5 )
     => ~ ! [X6: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P5
              = ( produc5700946648718959541nt_int @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7524_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P5: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P5 )
     => ~ ! [X6: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P5
              = ( produc4305682042979456191nt_int @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7525_case__prodE,axiom,
    ! [C: int > int > $o,P5: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P5 )
     => ~ ! [X6: int,Y5: int] :
            ( ( P5
              = ( product_Pair_int_int @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7526_case__prodD,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7527_case__prodD,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7528_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7529_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7530_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_7531_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P5: product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P5 @ Z )
     => ~ ! [X6: nat,Y5: nat] :
            ( ( P5
              = ( product_Pair_nat_nat @ X6 @ Y5 ) )
           => ~ ( C @ X6 @ Y5 @ Z ) ) ) ).

% case_prodE'
thf(fact_7532_case__prodD_H,axiom,
    ! [R4: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R4 @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R4 @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_7533_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_7534_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_7535_not__exp__less__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_7536_exp__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_gt_zero
thf(fact_7537_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X6: real] :
          ( ( exp_real @ X6 )
          = Y ) ) ).

% exp_total
thf(fact_7538_exp__ge__zero,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).

% exp_ge_zero
thf(fact_7539_not__exp__le__zero,axiom,
    ! [X: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_7540_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ ( suc @ X6 ) @ A2 )
           => ( ( F @ ( suc @ X6 ) )
              = ( G @ ( suc @ X6 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7541_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ ( suc @ X6 ) @ A2 )
           => ( ( F @ ( suc @ X6 ) )
              = ( G @ ( suc @ X6 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7542_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_7543_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_7544_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_7545_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_7546_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).

% AND_upper1
thf(fact_7547_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).

% AND_lower
thf(fact_7548_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_ri631733984087533419it_int @ N @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_7549_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_7550_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).

% take_bit_nonnegative
thf(fact_7551_not__take__bit__negative,axiom,
    ! [N: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_7552_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_7553_signed__take__bit__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_7554_mult__exp__exp,axiom,
    ! [X: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7555_mult__exp__exp,axiom,
    ! [X: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7556_exp__add__commuting,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7557_exp__add__commuting,axiom,
    ! [X: real,Y: real] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
        = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7558_exp__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).

% exp_diff
thf(fact_7559_exp__diff,axiom,
    ! [X: real,Y: real] :
      ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
      = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).

% exp_diff
thf(fact_7560_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_7561_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_7562_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_7563_take__bit__set__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_7564_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_7565_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_7566_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X6 ) @ ( F @ X6 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X3: complex] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7567_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X6 ) @ ( F @ X6 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X3: real] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7568_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X6: set_nat] :
          ( ( member_set_nat @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X6 ) @ ( F @ X6 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X3: set_nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7569_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X6 ) @ ( F @ X6 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X3: int] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7570_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X6 ) @ ( F @ X6 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : ( minus_minus_nat @ ( F @ X3 ) @ ( G @ X3 ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7571_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7572_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7573_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7574_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ I5 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7575_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: int] :
                  ( ( member_int @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7576_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: complex] :
                  ( ( member_complex @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7577_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7578_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N ) )
     => ? [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X6 ) ) ) ) ).

% sum_SucD
thf(fact_7579_exp__gt__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).

% exp_gt_one
thf(fact_7580_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: int] :
              ( ( member_int @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: int] :
                  ( ( member_int @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7581_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: complex] :
              ( ( member_complex @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: complex] :
                  ( ( member_complex @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7582_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X3: nat] :
              ( ( member_nat @ X3 @ A2 )
              & ( ( F @ X3 )
                = one_one_nat )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A2 )
                 => ( ( X3 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7583_take__bit__signed__take__bit,axiom,
    ! [M: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_7584_exp__ge__add__one__self,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).

% exp_ge_add_one_self
thf(fact_7585_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_7586_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_7587_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_7588_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_7589_exp__minus__inverse,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_7590_exp__minus__inverse,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_7591_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_7592_sum__power__add,axiom,
    ! [X: complex,M: nat,I6: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I5: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_7593_sum__power__add,axiom,
    ! [X: rat,M: nat,I6: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I5: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_7594_sum__power__add,axiom,
    ! [X: int,M: nat,I6: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_7595_sum__power__add,axiom,
    ! [X: real,M: nat,I6: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I5 ) )
        @ I6 )
      = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).

% sum_power_add
thf(fact_7596_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7597_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I5 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7598_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_7599_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X3: complex] : X3
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_7600_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_7601_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_7602_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_7603_sum__diff__nat,axiom,
    ! [B3: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7604_sum__diff__nat,axiom,
    ! [B3: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7605_sum__diff__nat,axiom,
    ! [B3: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7606_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_7607_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_7608_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_7609_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_7610_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_7611_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_7612_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_7613_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7614_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7615_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7616_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7617_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X6 )
          & ( ord_less_eq_real @ X6 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X6 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_7618_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7619_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7620_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7621_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7622_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7623_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7624_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7625_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7626_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).

% ln_ge_iff
thf(fact_7627_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_7628_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7629_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7630_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7631_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7632_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I5: nat] : ( minus_minus_rat @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7633_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( minus_minus_int @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7634_sum__Suc__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( minus_minus_real @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7635_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_7636_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_7637_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N2: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_7638_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_7639_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_7640_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7641_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7642_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_7643_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7644_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7645_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_7646_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_7647_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_7648_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_7649_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_nat_def
thf(fact_7650_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_7651_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_7652_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > rat,P5: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7653_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > int,P5: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7654_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > nat,P5: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7655_sum_Oub__add__nat,axiom,
    ! [M: nat,N: nat,G: nat > real,P5: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P5 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P5 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7656_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_def
thf(fact_7657_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_7658_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X3: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) @ ( plus_plus_real @ ( exp_real @ X3 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7659_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X3: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X3 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7660_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_7661_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_7662_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_7663_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_7664_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q5: int,R5: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_7665_take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_7666_take__bit__numeral__bit0,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_7667_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_7668_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_7669_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_7670_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_7671_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_7672_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_7673_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_7674_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_7675_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_7676_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_7677_sum__natinterval__diff,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_7678_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7679_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7680_sum__telescope_H_H,axiom,
    ! [M: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7681_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_7682_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_7683_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_7684_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_7685_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).

% divmod_nat_def
thf(fact_7686_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_7687_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
      = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7688_mask__eq__sum__exp,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).

% mask_eq_sum_exp
thf(fact_7689_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7690_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7691_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7692_sum__gp__multiplied,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7693_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I5: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_7694_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_7695_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_7696_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N: 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 ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% sum.in_pairs
thf(fact_7697_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_7698_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_7699_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_7700_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_7701_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_7702_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_7703_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_7704_take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_7705_take__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_7706_exp__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_7707_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_7708_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_7709_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_7710_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_7711_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_7712_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_7713_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_7714_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_7715_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_7716_take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_7717_take__bit__numeral__bit1,axiom,
    ! [L2: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_7718_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_7719_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_7720_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I5 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_7721_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or1269000886237332187st_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_7722_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_7723_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_7724_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_7725_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ 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 ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_7726_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_7727_sum__gp,axiom,
    ! [N: nat,M: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7728_sum__gp,axiom,
    ! [N: nat,M: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7729_sum__gp,axiom,
    ! [N: nat,M: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7730_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_7731_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = ( semiri8010041392384452111omplex @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_7732_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_7733_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = ( semiri681578069525770553at_rat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_7734_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_7735_of__nat__eq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M = N ) ) ).

% of_nat_eq_iff
thf(fact_7736_insert__subset,axiom,
    ! [X: complex,A2: set_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( insert_complex @ X @ A2 ) @ B3 )
      = ( ( member_complex @ X @ B3 )
        & ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_7737_insert__subset,axiom,
    ! [X: real,A2: set_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A2 ) @ B3 )
      = ( ( member_real @ X @ B3 )
        & ( ord_less_eq_set_real @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_7738_insert__subset,axiom,
    ! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
      = ( ( member_set_nat @ X @ B3 )
        & ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_7739_insert__subset,axiom,
    ! [X: nat,A2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
      = ( ( member_nat @ X @ B3 )
        & ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_7740_insert__subset,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X @ A2 ) @ B3 )
      = ( ( member_int @ X @ B3 )
        & ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% insert_subset
thf(fact_7741_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_7742_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_7743_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_7744_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_7745_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_7746_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7747_Diff__insert0,axiom,
    ! [X: complex,A2: set_complex,B3: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ B3 ) )
        = ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_7748_Diff__insert0,axiom,
    ! [X: real,A2: set_real,B3: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X @ B3 ) )
        = ( minus_minus_set_real @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_7749_Diff__insert0,axiom,
    ! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
        = ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_7750_Diff__insert0,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X @ B3 ) )
        = ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_7751_Diff__insert0,axiom,
    ! [X: nat,A2: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
        = ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_7752_insert__Diff1,axiom,
    ! [X: complex,B3: set_complex,A2: set_complex] :
      ( ( member_complex @ X @ B3 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B3 )
        = ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_7753_insert__Diff1,axiom,
    ! [X: real,B3: set_real,A2: set_real] :
      ( ( member_real @ X @ B3 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B3 )
        = ( minus_minus_set_real @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_7754_insert__Diff1,axiom,
    ! [X: set_nat,B3: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X @ B3 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
        = ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_7755_insert__Diff1,axiom,
    ! [X: int,B3: set_int,A2: set_int] :
      ( ( member_int @ X @ B3 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
        = ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_7756_insert__Diff1,axiom,
    ! [X: nat,B3: set_nat,A2: set_nat] :
      ( ( member_nat @ X @ B3 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
        = ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_7757_negative__zle,axiom,
    ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_7758_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7759_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_7760_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_7761_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_7762_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_7763_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_7764_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_7765_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_7766_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_7767_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_7768_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_7769_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_7770_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_7771_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_7772_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_7773_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_7774_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_7775_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_7776_of__nat__less__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_iff
thf(fact_7777_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_7778_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_7779_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_7780_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_7781_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_7782_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_7783_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_7784_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_7785_of__nat__le__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% of_nat_le_iff
thf(fact_7786_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_7787_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_7788_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_7789_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_7790_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_7791_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_7792_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_add
thf(fact_7793_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_7794_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_7795_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_7796_of__nat__add,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_7797_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% of_nat_mult
thf(fact_7798_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_7799_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_7800_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_7801_of__nat__mult,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_7802_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_7803_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_7804_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_7805_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_7806_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_7807_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7808_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7809_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7810_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7811_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7812_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7813_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri5074537144036343181t_real @ N )
        = one_one_real )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7814_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri681578069525770553at_rat @ N )
        = one_one_rat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7815_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N )
        = one_one_nat )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7816_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri1314217659103216013at_int @ N )
        = one_one_int )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7817_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_7818_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_7819_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_7820_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_7821_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_7822_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_7823_insert__Diff__single,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_7824_insert__Diff__single,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_7825_insert__Diff__single,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_7826_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B3: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B3 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_7827_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B3: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_7828_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B3: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B3 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_7829_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B3: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_7830_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).

% of_nat_power
thf(fact_7831_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).

% of_nat_power
thf(fact_7832_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
      = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_7833_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).

% of_nat_power
thf(fact_7834_of__nat__power,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).

% of_nat_power
thf(fact_7835_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7836_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7837_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
        = ( semiri681578069525770553at_rat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7838_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7839_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X ) )
      = ( ( power_power_nat @ B @ W )
        = X ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7840_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7841_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7842_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri681578069525770553at_rat @ X )
        = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7843_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7844_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7845_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_7846_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_7847_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n1201886186963655149omplex @ P ) ) ).

% of_nat_of_bool
thf(fact_7848_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_7849_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_nat_of_bool
thf(fact_7850_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_7851_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_7852_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_7853_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_7854_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_7855_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_7856_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_7857_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_7858_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_7859_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_7860_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_7861_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7862_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7863_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_7864_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7865_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7866_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7867_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7868_sum_Oinsert,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7869_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7870_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7871_sum_Oinsert,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7872_sum_Oinsert,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7873_sum_Oinsert,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_7874_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_7875_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_7876_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_7877_subset__Compl__singleton,axiom,
    ! [A2: set_complex,B: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
      = ( ~ ( member_complex @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7878_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7879_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7880_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7881_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_7882_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_7883_log__less__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_7884_log__less__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_real @ X @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_7885_one__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_real @ A @ X ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_7886_log__less__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_real @ X @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_7887_zero__less__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_real @ one_one_real @ X ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_7888_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_7889_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_7890_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_7891_of__nat__0__less__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% of_nat_0_less_iff
thf(fact_7892_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7893_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7894_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7895_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7896_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7897_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7898_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7899_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7900_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7901_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7902_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7903_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7904_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7905_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7906_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7907_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7908_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7909_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7910_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7911_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7912_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7913_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7914_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7915_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7916_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7917_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7918_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_7919_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_7920_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_7921_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_7922_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_7923_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_7924_log__le__cancel__iff,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_7925_log__le__one__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
          = ( ord_less_eq_real @ X @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_7926_one__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ A @ X ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_7927_log__le__zero__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
          = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_7928_zero__le__log__cancel__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
          = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_7929_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7930_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7931_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7932_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7933_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_7934_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7935_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7936_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7937_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7938_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_7939_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( numeral_numeral_real @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7940_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7941_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7942_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7943_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_7944_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7945_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7946_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7947_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7948_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_7949_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_7950_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7951_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7952_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7953_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7954_Suc__0__and__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_7955_and__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_7956_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7957_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7958_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7959_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7960_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7961_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7962_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7963_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7964_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_7965_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_7966_even__of__nat,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% even_of_nat
thf(fact_7967_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7968_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7969_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7970_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7971_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7972_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7973_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7974_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7975_Collect__case__prod__mono,axiom,
    ! [A2: int > int > $o,B3: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A2 @ B3 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B3 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_7976_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_7977_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_7978_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7979_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7980_mult__of__nat__commute,axiom,
    ! [X: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7981_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7982_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7983_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7984_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_7985_insert__mono,axiom,
    ! [C4: set_nat,D4: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C4 @ D4 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7986_insert__mono,axiom,
    ! [C4: set_real,D4: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C4 @ D4 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7987_insert__mono,axiom,
    ! [C4: set_int,D4: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C4 @ D4 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D4 ) ) ) ).

% insert_mono
thf(fact_7988_subset__insert,axiom,
    ! [X: complex,A2: set_complex,B3: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B3 ) )
        = ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_7989_subset__insert,axiom,
    ! [X: real,A2: set_real,B3: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B3 ) )
        = ( ord_less_eq_set_real @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_7990_subset__insert,axiom,
    ! [X: set_nat,A2: set_set_nat,B3: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_7991_subset__insert,axiom,
    ! [X: nat,A2: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
        = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_7992_subset__insert,axiom,
    ! [X: int,A2: set_int,B3: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) )
        = ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).

% subset_insert
thf(fact_7993_subset__insertI,axiom,
    ! [B3: set_nat,A: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat @ A @ B3 ) ) ).

% subset_insertI
thf(fact_7994_subset__insertI,axiom,
    ! [B3: set_real,A: real] : ( ord_less_eq_set_real @ B3 @ ( insert_real @ A @ B3 ) ) ).

% subset_insertI
thf(fact_7995_subset__insertI,axiom,
    ! [B3: set_int,A: int] : ( ord_less_eq_set_int @ B3 @ ( insert_int @ A @ B3 ) ) ).

% subset_insertI
thf(fact_7996_subset__insertI2,axiom,
    ! [A2: set_nat,B3: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_7997_subset__insertI2,axiom,
    ! [A2: set_real,B3: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B3 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_7998_subset__insertI2,axiom,
    ! [A2: set_int,B3: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_7999_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_8000_take__bit__of__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).

% take_bit_of_nat
thf(fact_8001_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_and_eq
thf(fact_8002_of__nat__and__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N ) )
      = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_and_eq
thf(fact_8003_insert__Diff__if,axiom,
    ! [X: complex,B3: set_complex,A2: set_complex] :
      ( ( ( member_complex @ X @ B3 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B3 )
          = ( minus_811609699411566653omplex @ A2 @ B3 ) ) )
      & ( ~ ( member_complex @ X @ B3 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ B3 )
          = ( insert_complex @ X @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_8004_insert__Diff__if,axiom,
    ! [X: real,B3: set_real,A2: set_real] :
      ( ( ( member_real @ X @ B3 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B3 )
          = ( minus_minus_set_real @ A2 @ B3 ) ) )
      & ( ~ ( member_real @ X @ B3 )
       => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ B3 )
          = ( insert_real @ X @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_8005_insert__Diff__if,axiom,
    ! [X: set_nat,B3: set_set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X @ B3 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
          = ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) )
      & ( ~ ( member_set_nat @ X @ B3 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ B3 )
          = ( insert_set_nat @ X @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_8006_insert__Diff__if,axiom,
    ! [X: int,B3: set_int,A2: set_int] :
      ( ( ( member_int @ X @ B3 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
          = ( minus_minus_set_int @ A2 @ B3 ) ) )
      & ( ~ ( member_int @ X @ B3 )
       => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ B3 )
          = ( insert_int @ X @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_8007_insert__Diff__if,axiom,
    ! [X: nat,B3: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X @ B3 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
          = ( minus_minus_set_nat @ A2 @ B3 ) ) )
      & ( ~ ( member_nat @ X @ B3 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B3 )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_8008_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N3: nat] :
          ( Z
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_diff_cases
thf(fact_8009_log__of__power__eq,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_8010_less__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_8011_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8012_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8013_of__nat__less__of__int__iff,axiom,
    ! [N: nat,X: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).

% of_nat_less_of_int_iff
thf(fact_8014_num__induct,axiom,
    ! [P: num > $o,X: num] :
      ( ( P @ one )
     => ( ! [X6: num] :
            ( ( P @ X6 )
           => ( P @ ( inc @ X6 ) ) )
       => ( P @ X ) ) ) ).

% num_induct
thf(fact_8015_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_8016_le__log__of__power,axiom,
    ! [B: real,N: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_8017_log__base__pow,axiom,
    ! [A: real,N: nat,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log_base_pow
thf(fact_8018_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ B @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_8019_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).

% of_nat_0_le_iff
thf(fact_8020_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_8021_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).

% of_nat_0_le_iff
thf(fact_8022_of__nat__0__le__iff,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).

% of_nat_0_le_iff
thf(fact_8023_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_8024_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_8025_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_8026_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_8027_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_8028_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_8029_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_8030_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_8031_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_8032_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_8033_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_8034_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_8035_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_8036_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_8037_less__imp__of__nat__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_8038_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_8039_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_8040_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_8041_of__nat__less__imp__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% of_nat_less_imp_less
thf(fact_8042_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_8043_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_8044_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_8045_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_8046_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_8047_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_8048_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_8049_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_8050_of__nat__dvd__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( dvd_dvd_nat @ M @ N ) ) ).

% of_nat_dvd_iff
thf(fact_8051_subset__singletonD,axiom,
    ! [A2: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_8052_subset__singletonD,axiom,
    ! [A2: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_8053_subset__singletonD,axiom,
    ! [A2: set_int,X: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_8054_subset__singleton__iff,axiom,
    ! [X8: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X8 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X8 = bot_bot_set_nat )
        | ( X8
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_8055_subset__singleton__iff,axiom,
    ! [X8: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X8 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X8 = bot_bot_set_real )
        | ( X8
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_8056_subset__singleton__iff,axiom,
    ! [X8: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X8 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X8 = bot_bot_set_int )
        | ( X8
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_8057_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_8058_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_8059_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_8060_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_8061_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_8062_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_8063_Diff__insert__absorb,axiom,
    ! [X: complex,A2: set_complex] :
      ( ~ ( member_complex @ X @ A2 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X @ A2 ) @ ( insert_complex @ X @ bot_bot_set_complex ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_8064_Diff__insert__absorb,axiom,
    ! [X: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A2 ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_8065_Diff__insert__absorb,axiom,
    ! [X: int,A2: set_int] :
      ( ~ ( member_int @ X @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_8066_Diff__insert__absorb,axiom,
    ! [X: real,A2: set_real] :
      ( ~ ( member_real @ X @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_8067_Diff__insert__absorb,axiom,
    ! [X: nat,A2: set_nat] :
      ( ~ ( member_nat @ X @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_8068_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B3: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_8069_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B3: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B3 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_8070_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_8071_insert__Diff,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( member_complex @ A @ A2 )
     => ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_8072_insert__Diff,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_8073_insert__Diff,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_8074_insert__Diff,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_8075_insert__Diff,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_8076_Diff__insert,axiom,
    ! [A2: set_int,A: int,B3: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B3 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_8077_Diff__insert,axiom,
    ! [A2: set_real,A: real,B3: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B3 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B3 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_8078_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_8079_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% zle_int
thf(fact_8080_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_8081_subset__Diff__insert,axiom,
    ! [A2: set_complex,B3: set_complex,X: complex,C4: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B3 @ ( insert_complex @ X @ C4 ) ) )
      = ( ( ord_le211207098394363844omplex @ A2 @ ( minus_811609699411566653omplex @ B3 @ C4 ) )
        & ~ ( member_complex @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_8082_subset__Diff__insert,axiom,
    ! [A2: set_real,B3: set_real,X: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B3 @ ( insert_real @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B3 @ C4 ) )
        & ~ ( member_real @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_8083_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B3: set_set_nat,X: set_nat,C4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B3 @ ( insert_set_nat @ X @ C4 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B3 @ C4 ) )
        & ~ ( member_set_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_8084_subset__Diff__insert,axiom,
    ! [A2: set_nat,B3: set_nat,X: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ C4 ) )
        & ~ ( member_nat @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_8085_subset__Diff__insert,axiom,
    ! [A2: set_int,B3: set_int,X: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ ( insert_int @ X @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B3 @ C4 ) )
        & ~ ( member_int @ X @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_8086_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% nonneg_int_cases
thf(fact_8087_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_8088_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).

% of_nat_mod
thf(fact_8089_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mod
thf(fact_8090_of__nat__mod,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mod
thf(fact_8091_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_8092_int__plus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_8093_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_8094_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_8095_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_8096_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z3: int] :
        ? [N2: nat] :
          ( Z3
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_8097_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_8098_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y ) ) ) ).

% of_nat_max
thf(fact_8099_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X ) @ ( semiri4939895301339042750nteger @ Y ) ) ) ).

% of_nat_max
thf(fact_8100_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_8101_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y ) ) ) ).

% of_nat_max
thf(fact_8102_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_8103_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_8104_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_8105_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_8106_log__of__power__less,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_8107_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_8108_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_8109_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% of_nat_mask_eq
thf(fact_8110_of__nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( bit_se2000444600071755411sk_int @ N ) ) ).

% of_nat_mask_eq
thf(fact_8111_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_8112_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_8113_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_8114_log__of__power__le,axiom,
    ! [M: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
     => ( ( 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 @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_8115_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_8116_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_8117_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_8118_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_8119_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).

% of_nat_diff
thf(fact_8120_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_8121_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_8122_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_8123_of__nat__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_8124_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_8125_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X6: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y2: complex] :
                    ( ( member_complex @ Y2 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8126_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X6: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y2: nat] :
                    ( ( member_nat @ Y2 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_nat @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8127_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X6: int,S5: set_int] :
              ( ( finite_finite_int @ S5 )
             => ( ! [Y2: int] :
                    ( ( member_int @ Y2 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_int @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8128_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X6: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y2: real] :
                    ( ( member_real @ Y2 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_real @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8129_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X6: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y2: complex] :
                    ( ( member_complex @ Y2 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8130_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X6: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y2: nat] :
                    ( ( member_nat @ Y2 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_nat @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8131_finite__ranking__induct,axiom,
    ! [S2: set_int,P: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X6: int,S5: set_int] :
              ( ( finite_finite_int @ S5 )
             => ( ! [Y2: int] :
                    ( ( member_int @ Y2 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_int @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8132_finite__ranking__induct,axiom,
    ! [S2: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X6: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y2: real] :
                    ( ( member_real @ Y2 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_real @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8133_finite__ranking__induct,axiom,
    ! [S2: set_complex,P: set_complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X6: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y2: complex] :
                    ( ( member_complex @ Y2 @ S5 )
                   => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8134_finite__ranking__induct,axiom,
    ! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X6: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y2: nat] :
                    ( ( member_nat @ Y2 @ S5 )
                   => ( ord_less_eq_nat @ ( F @ Y2 ) @ ( F @ X6 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_nat @ X6 @ S5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8135_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A7 )
                   => ( ord_less_real @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8136_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A7 )
                   => ( ord_less_rat @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8137_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A7 )
                   => ( ord_less_num @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8138_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                   => ( ord_less_nat @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8139_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A7 )
                   => ( ord_less_int @ X4 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8140_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B5: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A7 )
                   => ( ord_less_real @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8141_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B5: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A7 )
                   => ( ord_less_rat @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8142_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B5: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A7 )
                   => ( ord_less_num @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8143_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A7 )
                   => ( ord_less_nat @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8144_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A7 )
                   => ( ord_less_int @ B5 @ X4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B5 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8145_exp__of__nat2__mult,axiom,
    ! [X: complex,N: nat] :
      ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N ) ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_8146_exp__of__nat2__mult,axiom,
    ! [X: real,N: nat] :
      ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).

% exp_of_nat2_mult
thf(fact_8147_exp__of__nat__mult,axiom,
    ! [N: nat,X: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ X ) )
      = ( power_power_complex @ ( exp_complex @ X ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_8148_exp__of__nat__mult,axiom,
    ! [N: nat,X: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) )
      = ( power_power_real @ ( exp_real @ X ) @ N ) ) ).

% exp_of_nat_mult
thf(fact_8149_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8150_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8151_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8152_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8153_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8154_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8155_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8156_sum_Oinsert__if,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8157_sum_Oinsert__if,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8158_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
            = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_8159_finite__subset__induct,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ~ ( member_set_nat @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8160_finite__subset__induct,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ~ ( member_complex @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8161_finite__subset__induct,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ~ ( member_nat @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8162_finite__subset__induct,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ~ ( member_real @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8163_finite__subset__induct,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ~ ( member_int @ A5 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ A5 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_8164_finite__subset__induct_H,axiom,
    ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_set_nat )
         => ( ! [A5: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( member_set_nat @ A5 @ A2 )
                 => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
                   => ( ~ ( member_set_nat @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_set_nat @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8165_finite__subset__induct_H,axiom,
    ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A5: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( member_complex @ A5 @ A2 )
                 => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
                   => ( ~ ( member_complex @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_complex @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8166_finite__subset__induct_H,axiom,
    ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ord_less_eq_set_nat @ F3 @ A2 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A5: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat @ A5 @ A2 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A2 )
                   => ( ~ ( member_nat @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8167_finite__subset__induct_H,axiom,
    ! [F3: set_real,A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( ord_less_eq_set_real @ F3 @ A2 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A5: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( member_real @ A5 @ A2 )
                 => ( ( ord_less_eq_set_real @ F4 @ A2 )
                   => ( ~ ( member_real @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_real @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8168_finite__subset__induct_H,axiom,
    ! [F3: set_int,A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( ord_less_eq_set_int @ F3 @ A2 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A5: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( member_int @ A5 @ A2 )
                 => ( ( ord_less_eq_set_int @ F4 @ A2 )
                   => ( ~ ( member_int @ A5 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_int @ A5 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_8169_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y2: real] :
        ? [N3: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_8170_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_8171_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_8172_infinite__remove,axiom,
    ! [S2: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S2 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_8173_infinite__remove,axiom,
    ! [S2: set_int,A: int] :
      ( ~ ( finite_finite_int @ S2 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_8174_infinite__remove,axiom,
    ! [S2: set_real,A: real] :
      ( ~ ( finite_finite_real @ S2 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_8175_infinite__remove,axiom,
    ! [S2: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_8176_infinite__coinduct,axiom,
    ! [X8: set_complex > $o,A2: set_complex] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_complex] :
            ( ( X8 @ A7 )
           => ? [X4: complex] :
                ( ( member_complex @ X4 @ A7 )
                & ( ( X8 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_8177_infinite__coinduct,axiom,
    ! [X8: set_int > $o,A2: set_int] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_int] :
            ( ( X8 @ A7 )
           => ? [X4: int] :
                ( ( member_int @ X4 @ A7 )
                & ( ( X8 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_8178_infinite__coinduct,axiom,
    ! [X8: set_real > $o,A2: set_real] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_real] :
            ( ( X8 @ A7 )
           => ? [X4: real] :
                ( ( member_real @ X4 @ A7 )
                & ( ( X8 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_8179_infinite__coinduct,axiom,
    ! [X8: set_nat > $o,A2: set_nat] :
      ( ( X8 @ A2 )
     => ( ! [A7: set_nat] :
            ( ( X8 @ A7 )
           => ? [X4: nat] :
                ( ( member_nat @ X4 @ A7 )
                & ( ( X8 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_8180_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: set_nat,A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( member_set_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ A5 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_8181_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: complex,A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( member_complex @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A5 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_8182_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A5 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_8183_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( member_real @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A5 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_8184_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A5 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_8185_Diff__single__insert,axiom,
    ! [A2: set_real,X: real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B3 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_8186_Diff__single__insert,axiom,
    ! [A2: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_8187_Diff__single__insert,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_8188_subset__insert__iff,axiom,
    ! [A2: set_complex,X: complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ ( insert_complex @ X @ B3 ) )
      = ( ( ( member_complex @ X @ A2 )
         => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B3 ) )
        & ( ~ ( member_complex @ X @ A2 )
         => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_8189_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
      = ( ( ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B3 ) )
        & ( ~ ( member_set_nat @ X @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_8190_subset__insert__iff,axiom,
    ! [A2: set_real,X: real,B3: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X @ B3 ) )
      = ( ( ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B3 ) )
        & ( ~ ( member_real @ X @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_8191_subset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
      = ( ( ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 ) )
        & ( ~ ( member_nat @ X @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_8192_subset__insert__iff,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X @ B3 ) )
      = ( ( ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 ) )
        & ( ~ ( member_int @ X @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_8193_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% real_of_nat_div4
thf(fact_8194_sum__diff1__nat,axiom,
    ! [A: complex,A2: set_complex,F: complex > nat] :
      ( ( ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_complex @ A @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8195_sum__diff1__nat,axiom,
    ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
      ( ( ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_set_nat @ A @ A2 )
       => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
          = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8196_sum__diff1__nat,axiom,
    ! [A: int,A2: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8197_sum__diff1__nat,axiom,
    ! [A: real,A2: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A2 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8198_sum__diff1__nat,axiom,
    ! [A: nat,A2: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8199_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_8200_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_8201_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z3: int] :
        ? [N2: nat] :
          ( Z3
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_8202_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_8203_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( ( dvd_dvd_nat @ D @ N )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_8204_negative__zle__0,axiom,
    ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_8205_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% nonpos_int_cases
thf(fact_8206_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_8207_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X ) ) @ ( insert_real @ X @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_8208_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_8209_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X ) ) @ ( insert_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_8210_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X ) ) @ ( insert_int @ X @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_8211_Compl__insert,axiom,
    ! [X: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_8212_Compl__insert,axiom,
    ! [X: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_8213_Compl__insert,axiom,
    ! [X: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_8214_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_8215_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_8216_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_8217_log__base__change,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X )
          = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_8218_numeral__inc,axiom,
    ! [X: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_8219_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_real @ ( inc @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_8220_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_rat @ ( inc @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_8221_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_nat @ ( inc @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_8222_numeral__inc,axiom,
    ! [X: num] :
      ( ( numeral_numeral_int @ ( inc @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_8223_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_8224_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_8225_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_8226_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_8227_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_8228_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_8229_remove__induct,axiom,
    ! [P: set_set_nat > $o,B3: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B3 )
                 => ( ! [X4: set_nat] :
                        ( ( member_set_nat @ X4 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_8230_remove__induct,axiom,
    ! [P: set_complex > $o,B3: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B3 )
                 => ( ! [X4: complex] :
                        ( ( member_complex @ X4 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_8231_remove__induct,axiom,
    ! [P: set_real > $o,B3: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B3 )
                 => ( ! [X4: real] :
                        ( ( member_real @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_8232_remove__induct,axiom,
    ! [P: set_nat > $o,B3: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B3 )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_8233_remove__induct,axiom,
    ! [P: set_int > $o,B3: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B3 )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_8234_finite__remove__induct,axiom,
    ! [B3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B3 )
                 => ( ! [X4: set_nat] :
                        ( ( member_set_nat @ X4 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_8235_finite__remove__induct,axiom,
    ! [B3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B3 )
                 => ( ! [X4: complex] :
                        ( ( member_complex @ X4 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_8236_finite__remove__induct,axiom,
    ! [B3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B3 )
                 => ( ! [X4: real] :
                        ( ( member_real @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_8237_finite__remove__induct,axiom,
    ! [B3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B3 )
                 => ( ! [X4: nat] :
                        ( ( member_nat @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_8238_finite__remove__induct,axiom,
    ! [B3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B3 )
                 => ( ! [X4: int] :
                        ( ( member_int @ X4 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_8239_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_8240_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_8241_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_8242_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_8243_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_8244_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_8245_not__zle__0__negative,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).

% not_zle_0_negative
thf(fact_8246_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_8247_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N3: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_8248_finite__induct__select,axiom,
    ! [S2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T5: set_complex] :
              ( ( ord_less_set_complex @ T5 @ S2 )
             => ( ( P @ T5 )
               => ? [X4: complex] :
                    ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ S2 @ T5 ) )
                    & ( P @ ( insert_complex @ X4 @ T5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_8249_finite__induct__select,axiom,
    ! [S2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T5: set_int] :
              ( ( ord_less_set_int @ T5 @ S2 )
             => ( ( P @ T5 )
               => ? [X4: int] :
                    ( ( member_int @ X4 @ ( minus_minus_set_int @ S2 @ T5 ) )
                    & ( P @ ( insert_int @ X4 @ T5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_8250_finite__induct__select,axiom,
    ! [S2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T5: set_real] :
              ( ( ord_less_set_real @ T5 @ S2 )
             => ( ( P @ T5 )
               => ? [X4: real] :
                    ( ( member_real @ X4 @ ( minus_minus_set_real @ S2 @ T5 ) )
                    & ( P @ ( insert_real @ X4 @ T5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_8251_finite__induct__select,axiom,
    ! [S2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T5: set_nat] :
              ( ( ord_less_set_nat @ T5 @ S2 )
             => ( ( P @ T5 )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S2 @ T5 ) )
                    & ( P @ ( insert_nat @ X4 @ T5 ) ) ) ) )
         => ( P @ S2 ) ) ) ) ).

% finite_induct_select
thf(fact_8252_psubset__insert__iff,axiom,
    ! [A2: set_complex,X: complex,B3: set_complex] :
      ( ( ord_less_set_complex @ A2 @ ( insert_complex @ X @ B3 ) )
      = ( ( ( member_complex @ X @ B3 )
         => ( ord_less_set_complex @ A2 @ B3 ) )
        & ( ~ ( member_complex @ X @ B3 )
         => ( ( ( member_complex @ X @ A2 )
             => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) @ B3 ) )
            & ( ~ ( member_complex @ X @ A2 )
             => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_8253_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X: set_nat,B3: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X @ B3 ) )
      = ( ( ( member_set_nat @ X @ B3 )
         => ( ord_less_set_set_nat @ A2 @ B3 ) )
        & ( ~ ( member_set_nat @ X @ B3 )
         => ( ( ( member_set_nat @ X @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B3 ) )
            & ( ~ ( member_set_nat @ X @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_8254_psubset__insert__iff,axiom,
    ! [A2: set_real,X: real,B3: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X @ B3 ) )
      = ( ( ( member_real @ X @ B3 )
         => ( ord_less_set_real @ A2 @ B3 ) )
        & ( ~ ( member_real @ X @ B3 )
         => ( ( ( member_real @ X @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) @ B3 ) )
            & ( ~ ( member_real @ X @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_8255_psubset__insert__iff,axiom,
    ! [A2: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B3 ) )
      = ( ( ( member_nat @ X @ B3 )
         => ( ord_less_set_nat @ A2 @ B3 ) )
        & ( ~ ( member_nat @ X @ B3 )
         => ( ( ( member_nat @ X @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 ) )
            & ( ~ ( member_nat @ X @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_8256_psubset__insert__iff,axiom,
    ! [A2: set_int,X: int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X @ B3 ) )
      = ( ( ( member_int @ X @ B3 )
         => ( ord_less_set_int @ A2 @ B3 ) )
        & ( ~ ( member_int @ X @ B3 )
         => ( ( ( member_int @ X @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) @ B3 ) )
            & ( ~ ( member_int @ X @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_8257_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_8258_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_8259_set__replicate__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_8260_set__replicate__Suc,axiom,
    ! [N: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_8261_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_8262_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8263_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8264_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8265_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8266_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_8267_real__of__nat__div__aux,axiom,
    ! [X: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_8268_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_8269_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I5: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I5 ) @ bot_bot_set_int @ ( insert_int @ I5 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_8270_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_8271_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_8272_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8273_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_8274_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_8275_of__nat__less__two__power,axiom,
    ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% of_nat_less_two_power
thf(fact_8276_log__divide,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_8277_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_8278_inverse__of__nat__le,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_8279_exp__divide__power__eq,axiom,
    ! [N: nat,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
        = ( exp_complex @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_8280_exp__divide__power__eq,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
        = ( exp_real @ X ) ) ) ).

% exp_divide_power_eq
thf(fact_8281_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_8282_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8283_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8284_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8285_sum_Oremove,axiom,
    ! [A2: set_complex,X: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8286_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8287_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8288_sum_Oremove,axiom,
    ! [A2: set_int,X: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8289_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8290_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8291_sum_Oremove,axiom,
    ! [A2: set_real,X: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8292_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8293_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8294_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8295_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X @ A2 ) )
        = ( plus_plus_int @ ( G @ X ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8296_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > real,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8297_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > rat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8298_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > nat,X: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8299_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > real,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_real @ ( G @ X ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8300_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > rat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_rat @ ( G @ X ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8301_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X @ A2 ) )
        = ( plus_plus_nat @ ( G @ X ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8302_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8303_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8304_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8305_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8306_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8307_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8308_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8309_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8310_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ A @ A2 )
         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_nat @ A @ A2 )
         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8311_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ A @ A2 )
         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_nat @ A @ A2 )
         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( groups3539618377306564664at_int @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_8312_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_8313_zdiff__int__split,axiom,
    ! [P: int > $o,X: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_8314_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) ) ).

% real_of_nat_div2
thf(fact_8315_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_8316_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_8317_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8318_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K2: complex] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8319_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K2: complex] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8320_sum_Odelta__remove,axiom,
    ! [S2: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S2 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K2: complex] : ( if_int @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8321_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8322_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K2: int] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8323_sum_Odelta__remove,axiom,
    ! [S2: set_int,A: int,B: int > nat,C: int > nat] :
      ( ( finite_finite_int @ S2 )
     => ( ( ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S2 )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [K2: int] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups4541462559716669496nt_nat @ C @ ( minus_minus_set_int @ S2 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8324_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8325_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8326_sum_Odelta__remove,axiom,
    ! [S2: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S2 )
     => ( ( ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S2 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K2: real] : ( if_nat @ ( K2 = A ) @ ( B @ K2 ) @ ( C @ K2 ) )
              @ S2 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S2 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8327_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8328_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8329_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8330_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8331_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8332_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8333_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8334_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > rat] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X6 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8335_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > nat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X6: complex] :
            ( ( member_complex @ X6 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8336_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > nat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X6: int] :
            ( ( member_int @ X6 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8337_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > nat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X6 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_8338_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_8339_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_8340_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N2 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_8341_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_8342_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum
thf(fact_8343_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_8344_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum
thf(fact_8345_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum
thf(fact_8346_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum
thf(fact_8347_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8348_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I5: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8349_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8350_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8351_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I5 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_8352_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_8353_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_8354_arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I5 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8355_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I5 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_8356_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8357_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8358_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8359_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8360_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_8361_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_8362_sum__gp__offset,axiom,
    ! [X: complex,M: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8363_sum__gp__offset,axiom,
    ! [X: rat,M: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8364_sum__gp__offset,axiom,
    ! [X: real,M: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8365_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8366_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8367_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_8368_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8369_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_8370_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N2: nat] :
          ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M3: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_8371_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N2: nat] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M3: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_8372_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N2: nat] :
          ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M3: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_8373_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M3: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_8374_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] :
          ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M3: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_8375_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8376_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N: 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 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_8377_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N: 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 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_8378_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8379_lemma__termdiff2,axiom,
    ! [H2: complex,Z: complex,N: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H2
          @ ( groups2073611262835488442omplex
            @ ^ [P2: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P2 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8380_lemma__termdiff2,axiom,
    ! [H2: rat,Z: rat,N: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P2: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P2 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8381_lemma__termdiff2,axiom,
    ! [H2: real,Z: real,N: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H2
          @ ( groups6591440286371151544t_real
            @ ^ [P2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P2 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_8382_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_8383_lessThan__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( set_ord_lessThan_int @ X )
        = ( set_ord_lessThan_int @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_8384_lessThan__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( set_or5984915006950818249n_real @ X )
        = ( set_or5984915006950818249n_real @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_8385_lessThan__iff,axiom,
    ! [I: set_nat,K: set_nat] :
      ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
      = ( ord_less_set_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8386_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8387_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_8388_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_8389_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_8390_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_8391_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_8392_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8393_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8394_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8395_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8396_lessThan__subset__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8397_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8398_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8399_sum_OlessThan__Suc,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8400_sum_OlessThan__Suc,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8401_sum_OlessThan__Suc,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_8402_single__Diff__lessThan,axiom,
    ! [K: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
      = ( insert_nat @ K @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_8403_single__Diff__lessThan,axiom,
    ! [K: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
      = ( insert_int @ K @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_8404_single__Diff__lessThan,axiom,
    ! [K: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
      = ( insert_real @ K @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_8405_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8406_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_8407_set__encode__insert,axiom,
    ! [A2: set_nat,N: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ N @ A2 )
       => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).

% set_encode_insert
thf(fact_8408_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_8409_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_8410_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_8411_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_8412_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_8413_infinite__Iio,axiom,
    ! [A: real] :
      ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).

% infinite_Iio
thf(fact_8414_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X3: set_nat] : ( ord_less_set_nat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8415_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X3: rat] : ( ord_less_rat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8416_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X3: num] : ( ord_less_num @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8417_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8418_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X3: int] : ( ord_less_int @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8419_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X3: real] : ( ord_less_real @ X3 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_8420_Iio__eq__empty__iff,axiom,
    ! [N: extended_enat] :
      ( ( ( set_or8419480210114673929d_enat @ N )
        = bot_bo7653980558646680370d_enat )
      = ( N = bot_bo4199563552545308370d_enat ) ) ).

% Iio_eq_empty_iff
thf(fact_8421_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_8422_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8423_lessThan__strict__subset__iff,axiom,
    ! [M: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8424_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8425_lessThan__strict__subset__iff,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8426_lessThan__strict__subset__iff,axiom,
    ! [M: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_8427_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_8428_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_8429_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8430_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).

% finite_nat_bounded
thf(fact_8431_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8432_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_8433_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( set_or1269000886237332187st_nat @ M @ N )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_8434_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_8435_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_8436_norm__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).

% norm_exp
thf(fact_8437_norm__exp,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).

% norm_exp
thf(fact_8438_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8439_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_8440_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N: int] :
      ( ! [X6: int] : ( ord_less_eq_nat @ ( Q @ X6 ) @ ( P @ X6 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X3: int] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_ord_lessThan_int @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8441_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N: real] :
      ( ! [X6: real] : ( ord_less_eq_nat @ ( Q @ X6 ) @ ( P @ X6 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X3: real] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8442_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N: nat] :
      ( ! [X6: nat] : ( ord_less_eq_nat @ ( Q @ X6 ) @ ( P @ X6 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : ( minus_minus_nat @ ( P @ X3 ) @ ( Q @ X3 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8443_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8444_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8445_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8446_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_8447_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8448_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8449_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8450_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8451_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8452_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8453_sumr__diff__mult__const2,axiom,
    ! [F: nat > complex,N: nat,R: complex] :
      ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
      = ( groups2073611262835488442omplex
        @ ^ [I5: nat] : ( minus_minus_complex @ ( F @ I5 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8454_sumr__diff__mult__const2,axiom,
    ! [F: nat > rat,N: nat,R: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I5: nat] : ( minus_minus_rat @ ( F @ I5 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8455_sumr__diff__mult__const2,axiom,
    ! [F: nat > int,N: nat,R: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
      = ( groups3539618377306564664at_int
        @ ^ [I5: nat] : ( minus_minus_int @ ( F @ I5 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8456_sumr__diff__mult__const2,axiom,
    ! [F: nat > real,N: nat,R: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
      = ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( minus_minus_real @ ( F @ I5 ) @ R )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_8457_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8458_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8459_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8460_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8461_power__diff__1__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8462_power__diff__1__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_8463_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8464_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8465_one__diff__power__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8466_one__diff__power__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_8467_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_8468_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_8469_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_8470_monoseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).

% monoseq_realpow
thf(fact_8471_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri8010041392384452111omplex @ N ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8472_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8473_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri5074537144036343181t_real @ N ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_8474_lemma__termdiff1,axiom,
    ! [Z: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P2: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_complex @ Z @ P2 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P2: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P2 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P2 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8475_lemma__termdiff1,axiom,
    ! [Z: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P2: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_rat @ Z @ P2 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P2: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P2 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P2 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8476_lemma__termdiff1,axiom,
    ! [Z: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P2: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_int @ Z @ P2 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P2: nat] : ( times_times_int @ ( power_power_int @ Z @ P2 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P2 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8477_lemma__termdiff1,axiom,
    ! [Z: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P2: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_real @ Z @ P2 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P2: nat] : ( times_times_real @ ( power_power_real @ Z @ P2 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P2 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P2 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_8478_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) ) @ ( power_power_complex @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8479_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I5: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) ) @ ( power_power_rat @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8480_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) ) @ ( power_power_int @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8481_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) ) @ ( power_power_real @ X @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8482_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P2: nat] : ( times_times_complex @ ( power_power_complex @ X @ P2 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P2 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8483_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P2: nat] : ( times_times_rat @ ( power_power_rat @ X @ P2 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P2 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8484_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P2: nat] : ( times_times_int @ ( power_power_int @ X @ P2 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P2 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8485_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P2: nat] : ( times_times_real @ ( power_power_real @ X @ P2 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P2 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8486_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_8487_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K5: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( 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 @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8488_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K5: int,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( 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 @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8489_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K5: nat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( 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 @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8490_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K5: real,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( 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 @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8491_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_8492_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_8493_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I5: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8494_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I5: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8495_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I5: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8496_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I5 ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_8497_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ ( F @ I5 ) @ ( G @ I5 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I5: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I5 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_8498_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_8499_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_8500_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8501_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_8502_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_8503_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_8504_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_8505_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_8506_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_8507_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_8508_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_8509_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_8510_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_8511_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_8512_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_8513_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_8514_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_8515_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_8516_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_8517_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_8518_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_8519_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_8520_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_8521_norm__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
      = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_8522_norm__minus__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
      = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).

% norm_minus_commute
thf(fact_8523_norm__not__less__zero,axiom,
    ! [X: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_8524_norm__ge__zero,axiom,
    ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).

% norm_ge_zero
thf(fact_8525_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_8526_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_8527_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_8528_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_8529_sum__norm__le,axiom,
    ! [S2: set_real,F: real > complex,G: real > real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8530_sum__norm__le,axiom,
    ! [S2: set_set_nat,F: set_nat > complex,G: set_nat > real] :
      ( ! [X6: set_nat] :
          ( ( member_set_nat @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S2 ) ) @ ( groups5107569545109728110t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8531_sum__norm__le,axiom,
    ! [S2: set_int,F: int > complex,G: int > real] :
      ( ! [X6: int] :
          ( ( member_int @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8532_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8533_sum__norm__le,axiom,
    ! [S2: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X6: complex] :
          ( ( member_complex @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8534_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > real,G: nat > real] :
      ( ! [X6: nat] :
          ( ( member_nat @ X6 @ S2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X6 ) ) @ ( G @ X6 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8535_norm__power,axiom,
    ! [X: real,N: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power
thf(fact_8536_norm__power,axiom,
    ! [X: complex,N: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power
thf(fact_8537_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( real_V1022390504157884413omplex @ ( F @ I5 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8538_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I5: complex] : ( real_V1022390504157884413omplex @ ( F @ I5 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8539_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I5: nat] : ( real_V7735802525324610683m_real @ ( F @ I5 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8540_norm__uminus__minus,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_8541_norm__uminus__minus,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_8542_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_8543_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_8544_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z: real] :
      ( ( ( power_power_real @ W @ N )
        = ( power_power_real @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_8545_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N )
        = ( power_power_complex @ Z @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_8546_norm__mult__less,axiom,
    ! [X: real,R: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_8547_norm__mult__less,axiom,
    ! [X: complex,R: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_8548_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_8549_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_8550_norm__triangle__lt,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_8551_norm__triangle__lt,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_8552_norm__add__less,axiom,
    ! [X: real,R: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_8553_norm__add__less,axiom,
    ! [X: complex,R: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_8554_norm__power__ineq,axiom,
    ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_8555_norm__power__ineq,axiom,
    ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).

% norm_power_ineq
thf(fact_8556_norm__triangle__mono,axiom,
    ! [A: real,R: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_8557_norm__triangle__mono,axiom,
    ! [A: complex,R: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_8558_norm__triangle__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_8559_norm__triangle__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_8560_norm__triangle__le,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_8561_norm__triangle__le,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_8562_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_8563_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_8564_norm__diff__triangle__less,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_8565_norm__diff__triangle__less,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_8566_norm__triangle__le__diff,axiom,
    ! [X: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_8567_norm__triangle__le__diff,axiom,
    ! [X: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).

% norm_triangle_le_diff
thf(fact_8568_norm__diff__triangle__le,axiom,
    ! [X: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_8569_norm__diff__triangle__le,axiom,
    ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_8570_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_8571_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_8572_norm__triangle__sub,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_8573_norm__triangle__sub,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_8574_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_8575_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_8576_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_8577_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_8578_power__eq__1__iff,axiom,
    ! [W: real,N: nat] :
      ( ( ( power_power_real @ W @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_8579_power__eq__1__iff,axiom,
    ! [W: complex,N: nat] :
      ( ( ( power_power_complex @ W @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_8580_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_8581_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_8582_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_8583_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_8584_square__norm__one,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_8585_square__norm__one,axiom,
    ! [X: complex] :
      ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_8586_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_8587_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8588_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8589_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_8590_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: 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 @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_8591_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8592_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8593_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_8594_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_8595_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_8596_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_8597_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_8598_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_8599_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8600_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).

% summable_rabs
thf(fact_8601_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8602_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_8603_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_8604_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_8605_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_8606_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_8607_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
              @ ^ [I5: nat] : ( times_times_real @ ( F @ I5 ) @ ( power_power_real @ Z @ I5 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8608_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_8609_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_8610_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_8611_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8612_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_8613_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_8614_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_8615_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_8616_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_8617_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_8618_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_8619_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_8620_sin__cos__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% sin_cos_npi
thf(fact_8621_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_8622_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_8623_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_8624_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8625_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_8626_sin__pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
      = ( sin_real @ X ) ) ).

% sin_pi_minus
thf(fact_8627_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_8628_cos__periodic__pi2,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi2
thf(fact_8629_cos__periodic__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_periodic_pi
thf(fact_8630_sin__periodic__pi2,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi2
thf(fact_8631_sin__periodic__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_periodic_pi
thf(fact_8632_cos__minus__pi,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_minus_pi
thf(fact_8633_cos__pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
      = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).

% cos_pi_minus
thf(fact_8634_sin__minus__pi,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_minus_pi
thf(fact_8635_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8636_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8637_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8638_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_8639_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8640_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_8641_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8642_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_8643_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_8644_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_8645_cos__npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi
thf(fact_8646_cos__npi2,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).

% cos_npi2
thf(fact_8647_sin__2npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_8648_cos__2npi,axiom,
    ! [N: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_8649_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_8650_sin__int__2pin,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_8651_cos__int__2pin,axiom,
    ! [N: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_8652_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_8653_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_8654_cos__npi__int,axiom,
    ! [N: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_8655_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R2: real,A5: real] :
      ( ( X
        = ( times_times_real @ R2 @ ( cos_real @ A5 ) ) )
      & ( Y
        = ( times_times_real @ R2 @ ( sin_real @ A5 ) ) ) ) ).

% polar_Ex
thf(fact_8656_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_8657_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y5 )
      & ( ord_less_eq_real @ Y5 @ pi )
      & ( ( sin_real @ Y5 )
        = ( sin_real @ X ) )
      & ( ( cos_real @ Y5 )
        = ( cos_real @ X ) ) ) ).

% sincos_principal_value
thf(fact_8658_sin__x__le__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).

% sin_x_le_x
thf(fact_8659_sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).

% sin_le_one
thf(fact_8660_cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).

% cos_le_one
thf(fact_8661_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).

% abs_sin_x_le_abs_x
thf(fact_8662_cos__arctan__not__zero,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_8663_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_8664_sin__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero
thf(fact_8665_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).

% sin_x_ge_neg_x
thf(fact_8666_sin__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_ge_zero
thf(fact_8667_sin__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).

% sin_ge_minus_one
thf(fact_8668_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X )
                = ( cos_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_8669_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_8670_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_8671_cos__ge__minus__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).

% cos_ge_minus_one
thf(fact_8672_abs__sin__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_8673_abs__cos__le__one,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_8674_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_8675_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8676_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8677_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
     => ( ( ord_less_real @ X @ pi )
       => ( ( ( sin_real @ X )
            = zero_zero_real )
         => ( X = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_8678_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_8679_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_8680_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I5: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I5 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8681_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ pi )
            & ( X
              = ( cos_real @ T6 ) )
            & ( Y
              = ( sin_real @ T6 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_8682_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8683_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8684_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero_02
thf(fact_8685_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8686_cos__is__zero,axiom,
    ? [X6: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X6 )
      & ( ord_less_eq_real @ X6 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X6 )
        = zero_zero_real )
      & ! [Y2: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
            & ( ord_less_eq_real @ Y2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y2 )
              = zero_zero_real ) )
         => ( Y2 = X6 ) ) ) ).

% cos_is_zero
thf(fact_8687_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_8688_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8689_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X6 )
            & ( ord_less_eq_real @ X6 @ pi )
            & ( ( cos_real @ X6 )
              = Y )
            & ! [Y2: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
                  & ( ord_less_eq_real @ Y2 @ pi )
                  & ( ( cos_real @ Y2 )
                    = Y ) )
               => ( Y2 = X6 ) ) ) ) ) ).

% cos_total
thf(fact_8690_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T6: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X
                = ( cos_real @ T6 ) )
              & ( Y
                = ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_8691_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8692_square__fact__le__2__fact,axiom,
    ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% square_fact_le_2_fact
thf(fact_8693_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8694_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8695_sin__gt__zero2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).

% sin_gt_zero2
thf(fact_8696_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8697_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8698_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_8699_cos__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero
thf(fact_8700_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_8701_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_8702_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X )
                = ( sin_real @ Y ) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_8703_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_8704_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X3: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8705_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8706_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8707_sin__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8708_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8709_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8710_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X6: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X6 )
            & ( ord_less_eq_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X6 )
              = Y )
            & ! [Y2: real] :
                ( ( ( 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 @ Y2 )
                    = Y ) )
               => ( Y2 = X6 ) ) ) ) ) ).

% sin_total
thf(fact_8711_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_8712_cos__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).

% cos_ge_zero
thf(fact_8713_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X3: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X3: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8714_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 ) )
     => ( ! [K3: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
           => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K3 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_8715_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B8: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8716_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8717_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8718_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_8719_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8720_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I5: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I5 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I5 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8721_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I5: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I5 )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I5 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8722_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8723_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8724_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8725_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8726_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N2: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8727_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8728_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8729_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8730_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8731_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8732_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8733_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_8734_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_8735_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8736_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_Suc_0_nat
thf(fact_8737_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_8738_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8739_fact__div__fact__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8740_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_8741_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8742_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8743_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_8744_tan__periodic__pi,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_pi
thf(fact_8745_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8746_tan__periodic__n,axiom,
    ! [X: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_8747_tan__periodic__nat,axiom,
    ! [X: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_8748_tan__periodic__int,axiom,
    ! [X: real,I: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_8749_norm__cos__sin,axiom,
    ! [T: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_8750_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_8751_complex__diff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
      = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).

% complex_diff
thf(fact_8752_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_8753_Complex__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_8754_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_8755_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_8756_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_8757_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_8758_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_8759_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_8760_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_8761_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_8762_power__half__series,axiom,
    ( sums_real
    @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8763_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X6: real] :
          ( ( ord_less_real @ zero_zero_real @ X6 )
          & ( ord_less_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X6 ) ) ) ) ).

% lemma_tan_total
thf(fact_8764_tan__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_gt_zero
thf(fact_8765_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X6: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X6 )
      & ( ord_less_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X6 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_8766_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_8767_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
         => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_8768_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X )
       => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).

% tan_monotone
thf(fact_8769_tan__total,axiom,
    ! [Y: real] :
    ? [X6: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X6 )
      & ( ord_less_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X6 )
        = Y )
      & ! [Y2: 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 ) ) ) )
            & ( ( tan_real @ Y2 )
              = Y ) )
         => ( Y2 = X6 ) ) ) ).

% tan_total
thf(fact_8770_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_8771_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_8772_sums__if_H,axiom,
    ! [G: nat > real,X: real] :
      ( ( sums_real @ G @ X )
     => ( sums_real
        @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X ) ) ).

% sums_if'
thf(fact_8773_sums__if,axiom,
    ! [G: nat > real,X: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sums_if
thf(fact_8774_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8775_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X6 )
          & ( ord_less_real @ X6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X6 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_8776_tan__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_8777_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8778_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_8779_tan__bound__pi2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8780_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X )
            = Y )
         => ( ( arctan @ Y )
            = X ) ) ) ) ).

% arctan_unique
thf(fact_8781_arctan__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X ) )
          = X ) ) ) ).

% arctan_tan
thf(fact_8782_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_8783_tan__total__pi4,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ? [Z5: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z5 )
          & ( ord_less_real @ Z5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z5 )
            = X ) ) ) ).

% tan_total_pi4
thf(fact_8784_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_8785_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
            & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8786_sin__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X )
        = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8787_real__sqrt__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ X )
        = ( sqrt @ Y ) )
      = ( X = Y ) ) ).

% real_sqrt_eq_iff
thf(fact_8788_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_8789_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_8790_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_8791_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_8792_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( sqrt @ X )
        = one_one_real )
      = ( X = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_8793_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_8794_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8795_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8796_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_8797_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_8798_powr__gt__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
      = ( X != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8799_powr__nonneg__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8800_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8801_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_real @ X @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8802_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
      = ( ord_less_eq_real @ X @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_8803_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_8804_powr__less__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_8805_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8806_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_8807_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_8808_powr__eq__one__iff,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X )
          = one_one_real )
        = ( X = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_8809_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( ( powr_real @ X @ one_one_real )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% powr_one_gt_zero_iff
thf(fact_8810_powr__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ one_one_real )
        = X ) ) ).

% powr_one
thf(fact_8811_powr__le__cancel__iff,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_8812_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8813_powr__log__cancel,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( powr_real @ A @ ( log @ A @ X ) )
            = X ) ) ) ) ).

% powr_log_cancel
thf(fact_8814_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_8815_real__sqrt__abs,axiom,
    ! [X: real] :
      ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs
thf(fact_8816_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
        = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).

% powr_numeral
thf(fact_8817_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X ) ) ).

% real_sqrt_pow2
thf(fact_8818_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% real_sqrt_pow2_iff
thf(fact_8819_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_8820_square__powr__half,axiom,
    ! [X: real] :
      ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X ) ) ).

% square_powr_half
thf(fact_8821_real__sqrt__minus,axiom,
    ! [X: real] :
      ( ( sqrt @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_minus
thf(fact_8822_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_8823_real__sqrt__power,axiom,
    ! [X: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X @ K ) )
      = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).

% real_sqrt_power
thf(fact_8824_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_8825_real__sqrt__divide,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_divide
thf(fact_8826_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_8827_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_8828_powr__non__neg,axiom,
    ! [A: real,X: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8829_powr__less__mono2__neg,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_8830_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).

% powr_ge_pzero
thf(fact_8831_powr__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_8832_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8833_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sqrt @ X )
          = zero_zero_real )
       => ( X = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_8834_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_zero
thf(fact_8835_powr__less__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_less_mono
thf(fact_8836_powr__less__cancel,axiom,
    ! [X: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
     => ( ( ord_less_real @ one_one_real @ X )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_8837_powr__mono,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X )
       => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).

% powr_mono
thf(fact_8838_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_ge_one
thf(fact_8839_powr__mono2_H,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_8840_powr__less__mono2,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ Y )
         => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_8841_powr__inj,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X )
            = ( powr_real @ A @ Y ) )
          = ( X = Y ) ) ) ) ).

% powr_inj
thf(fact_8842_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_8843_ge__one__powr__ge__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_8844_powr__mono__both,axiom,
    ! [A: real,B: real,X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ( ord_less_eq_real @ X @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_8845_powr__le1,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_8846_powr__divide,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_8847_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_8848_real__div__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
        = ( sqrt @ X ) ) ) ).

% real_div_sqrt
thf(fact_8849_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_8850_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8851_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_8852_log__base__powr,axiom,
    ! [A: real,B: real,X: real] :
      ( ( A != zero_zero_real )
     => ( ( log @ ( powr_real @ A @ B ) @ X )
        = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).

% log_base_powr
thf(fact_8853_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_8854_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).

% log_powr
thf(fact_8855_powr__half__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X ) ) ) ).

% powr_half_sqrt
thf(fact_8856_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8857_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
        = ( power_power_real @ X @ N ) ) ) ).

% powr_realpow
thf(fact_8858_powr__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_less_iff
thf(fact_8859_less__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_8860_log__less__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_8861_less__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% less_log_iff
thf(fact_8862_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_8863_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
     => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_8864_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_8865_powr__neg__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X ) ) ) ).

% powr_neg_one
thf(fact_8866_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_8867_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_8868_le__log__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).

% le_log_iff
thf(fact_8869_log__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
          = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_8870_le__powr__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_8871_powr__le__iff,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).

% powr_le_iff
thf(fact_8872_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_8873_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_8874_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_8875_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_8876_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_8877_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_8878_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_8879_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_8880_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_8881_ln__powr__bound,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_8882_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_8883_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_8884_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_8885_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8886_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8887_minus__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8888_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_8889_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8890_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_8891_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_8892_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8893_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8894_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8895_ln__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( sqrt @ X ) )
        = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8896_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X3: real] : ( ln_ln_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8897_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_8898_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_8899_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
            = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8900_complex__norm,axiom,
    ! [X: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_8901_arsinh__real__aux,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8902_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( sqrt @ X ) @ N )
          = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8903_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8904_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8905_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8906_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8907_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8908_arcosh__real__def,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ one_one_real @ X )
     => ( ( arcosh_real @ X )
        = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8909_cos__arctan,axiom,
    ! [X: real] :
      ( ( cos_real @ ( arctan @ X ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_8910_sin__arctan,axiom,
    ! [X: real] :
      ( ( sin_real @ ( arctan @ X ) )
      = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8911_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8912_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
     => ( ( sin_real @ X )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8913_arctan__half,axiom,
    ( arctan
    = ( ^ [X3: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X3 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8914_cos__tan,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8915_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( A5
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A5
                   => ( ( B5
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_8916_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A5: $o,B5: $o] :
              ( ( X
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( ( B5
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B5
                   => ( ( A5
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A5
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B5 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_8917_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A5: complex,R2: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( exp_complex @ A5 ) ) ) ).

% complex_exp_exists
thf(fact_8918_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( complex2 @ R5 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_8919_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X3: real] : ( complex2 @ X3 @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_8920_complex__eq__cancel__iff2,axiom,
    ! [X: real,Y: real,Xa2: real] :
      ( ( ( complex2 @ X @ Y )
        = ( real_V4546457046886955230omplex @ Xa2 ) )
      = ( ( X = Xa2 )
        & ( Y = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_8921_complex__of__real__mult__Complex,axiom,
    ! [R: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_8922_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y @ R ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_8923_complex__of__real__add__Complex,axiom,
    ! [R: real,X: real,Y: real] :
      ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).

% complex_of_real_add_Complex
thf(fact_8924_Complex__add__complex__of__real,axiom,
    ! [X: real,Y: real,R: real] :
      ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).

% Complex_add_complex_of_real
thf(fact_8925_cos__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8926_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8927_sin__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8928_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_8929_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_8930_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_8931_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_8932_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_8933_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8934_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8935_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_8936_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_8937_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X )
          = ( arccos @ Y ) )
        = ( X = Y ) ) ) ).

% arccos_eq_iff
thf(fact_8938_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% arccos_le_mono
thf(fact_8939_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_8940_arcsin__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X ) )
          = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).

% arcsin_minus
thf(fact_8941_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X )
            = ( arcsin @ Y ) )
          = ( X = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_8942_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_8943_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_8944_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8945_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% arccos_less_mono
thf(fact_8946_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_8947_arccos__cos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ pi )
       => ( ( arccos @ ( cos_real @ X ) )
          = X ) ) ) ).

% arccos_cos
thf(fact_8948_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8949_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_8950_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_8951_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_8952_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8953_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8954_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( sin_real @ ( arccos @ X ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8955_arccos__cos2,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( arccos @ ( cos_real @ X ) )
          = ( uminus_uminus_real @ X ) ) ) ) ).

% arccos_cos2
thf(fact_8956_arccos__minus,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X ) )
          = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).

% arccos_minus
thf(fact_8957_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8958_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_8959_arccos__minus__abs,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X ) )
        = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).

% arccos_minus_abs
thf(fact_8960_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8961_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8962_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8963_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8964_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_8965_arcsin__sin,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X ) )
          = X ) ) ) ).

% arcsin_sin
thf(fact_8966_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8967_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
              = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8968_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_8969_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_8970_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_8971_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_8972_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_8973_binomial__code,axiom,
    ( binomial
    = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8974_binomial__n__n,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ N )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_8975_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_8976_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8977_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ( binomial @ N @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K ) ) ).

% binomial_eq_0_iff
thf(fact_8978_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_8979_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8980_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_8981_complex__i__mult__minus,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% complex_i_mult_minus
thf(fact_8982_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_8983_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
      = ( ord_less_eq_nat @ K @ N ) ) ).

% zero_less_binomial_iff
thf(fact_8984_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_8985_divide__numeral__i,axiom,
    ! [Z: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_8986_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_8987_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_8988_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_8989_i__even__power,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).

% i_even_power
thf(fact_8990_choose__one,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ one_one_nat )
      = N ) ).

% choose_one
thf(fact_8991_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_8992_complex__i__not__one,axiom,
    imaginary_unit != one_one_complex ).

% complex_i_not_one
thf(fact_8993_complex__i__not__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( numera6690914467698888265omplex @ W ) ) ).

% complex_i_not_numeral
thf(fact_8994_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ N @ K )
     => ( ( binomial @ N @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8995_Suc__times__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).

% Suc_times_binomial
thf(fact_8996_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).

% Suc_times_binomial_eq
thf(fact_8997_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_8998_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_8999_binomial__le__pow,axiom,
    ! [R: nat,N: nat] :
      ( ( ord_less_eq_nat @ R @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).

% binomial_le_pow
thf(fact_9000_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W )
        = Z )
      = ( W
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).

% i_times_eq_iff
thf(fact_9001_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).

% zero_less_binomial
thf(fact_9002_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_9003_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9004_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9005_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9006_complex__i__not__neg__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_i_not_neg_numeral
thf(fact_9007_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9008_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( ( complex2 @ X @ Y )
        = imaginary_unit )
      = ( ( X = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9009_binomial__absorption,axiom,
    ! [K: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9010_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_9011_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_9012_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_9013_binomial__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_9014_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_9015_binomial__maximum,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9016_binomial__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9017_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_9018_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9019_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9020_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ N )
     => ( ( binomial @ N @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9021_i__complex__of__real,axiom,
    ! [R: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
      = ( complex2 @ zero_zero_real @ R ) ) ).

% i_complex_of_real
thf(fact_9022_complex__of__real__i,axiom,
    ! [R: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R ) ) ).

% complex_of_real_i
thf(fact_9023_Complex__eq,axiom,
    ( complex2
    = ( ^ [A3: real,B2: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).

% Complex_eq
thf(fact_9024_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9025_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9026_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9027_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_9028_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9029_choose__two,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9030_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R2: real,A5: real] :
      ( Z
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_9031_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_9032_cmod__complex__polar,axiom,
    ! [R: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R ) ) ).

% cmod_complex_polar
thf(fact_9033_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_9034_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_9035_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_9036_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9037_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_9038_csqrt__0,axiom,
    ( ( csqrt @ zero_zero_complex )
    = zero_zero_complex ) ).

% csqrt_0
thf(fact_9039_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
thf(fact_9040_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% csqrt_eq_1
thf(fact_9041_csqrt__1,axiom,
    ( ( csqrt @ one_one_complex )
    = one_one_complex ) ).

% csqrt_1
thf(fact_9042_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9043_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9044_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9045_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9046_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9047_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K2: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9048_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_9049_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9050_sum__choose__lower,axiom,
    ! [R: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_9051_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9052_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9053_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9054_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9055_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
        @ ( set_ord_atMost_nat @ R ) )
      = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).

% vandermonde
thf(fact_9056_of__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).

% of_real_sqrt
thf(fact_9057_choose__row__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% choose_row_sum
thf(fact_9058_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_9059_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_9060_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I5: nat] : ( times_times_nat @ ( A @ I5 ) @ ( power_power_nat @ X @ I5 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9061_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_9062_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_9063_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I5: nat] : ( times_times_nat @ I5 @ ( binomial @ N @ I5 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9064_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N2: nat] : N2 ) ) ).

% of_nat_id
thf(fact_9065_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_9066_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_9067_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_9068_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N: 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 @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_9069_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_9070_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9071_cis__pi,axiom,
    ( ( cis @ pi )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cis_pi
thf(fact_9072_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_9073_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9074_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_9075_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9076_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_9077_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_9078_real__sqrt__inverse,axiom,
    ! [X: real] :
      ( ( sqrt @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).

% real_sqrt_inverse
thf(fact_9079_cis__neq__zero,axiom,
    ! [A: real] :
      ( ( cis @ A )
     != zero_zero_complex ) ).

% cis_neq_zero
thf(fact_9080_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X3: real,Y3: real] : ( times_times_real @ X3 @ ( inverse_inverse_real @ Y3 ) ) ) ) ).

% divide_real_def
thf(fact_9081_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_9082_cis__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).

% cis_divide
thf(fact_9083_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_9084_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_9085_floor__eq,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq
thf(fact_9086_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_9087_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_9088_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_9089_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9090_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_9091_forall__pos__mono,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D3: real,E2: real] :
          ( ( ord_less_real @ D3 @ E2 )
         => ( ( P @ D3 )
           => ( P @ E2 ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_9092_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N2: nat] :
            ( ( N2 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_9093_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
        = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9094_ln__inverse,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).

% ln_inverse
thf(fact_9095_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X )
          = N ) ) ) ).

% floor_eq2
thf(fact_9096_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_9097_log__inverse,axiom,
    ! [A: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( log @ A @ ( inverse_inverse_real @ X ) )
            = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).

% log_inverse
thf(fact_9098_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B2: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B2 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_9099_exp__plus__inverse__exp,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9100_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9101_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9102_tan__cot,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).

% tan_cot
thf(fact_9103_real__le__x__sinh,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9104_real__le__abs__sinh,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9105_floor__log__eq__powr__iff,axiom,
    ! [X: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
            & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9106_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_9107_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9108_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9109_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9110_sinh__real__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sinh_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9111_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X @ Y ) ) ).

% sinh_real_le_iff
thf(fact_9112_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_real @ X @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9113_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_real @ zero_zero_real @ X ) ) ).

% sinh_real_pos_iff
thf(fact_9114_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9115_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% sinh_real_nonneg_iff
thf(fact_9116_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X3: complex,Y3: complex] : ( times_times_complex @ X3 @ ( invers8013647133539491842omplex @ Y3 ) ) ) ) ).

% divide_complex_def
thf(fact_9117_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X3: nat] : X3
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9118_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9119_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_9120_sinh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( sinh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9121_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9122_sinh__le__cosh__real,axiom,
    ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).

% sinh_le_cosh_real
thf(fact_9123_cosh__real__nonzero,axiom,
    ! [X: real] :
      ( ( cosh_real @ X )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9124_cosh__real__pos,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_pos
thf(fact_9125_cosh__real__nonneg,axiom,
    ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).

% cosh_real_nonneg
thf(fact_9126_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9127_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9128_cosh__real__ge__1,axiom,
    ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).

% cosh_real_ge_1
thf(fact_9129_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9130_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9131_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9132_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9133_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( arcosh_real @ ( cosh_real @ X ) )
        = X ) ) ).

% arcosh_cosh_real
thf(fact_9134_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X3: int] : X3
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9135_cosh__ln__real,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( cosh_real @ ( ln_ln_real @ X ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9136_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( 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_9137_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9138_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9139_cot__less__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9140_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z6 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9141_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D2: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D2 @ Z3 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9142_real__root__zero,axiom,
    ! [N: nat] :
      ( ( root @ N @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9143_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_9144_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9145_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_9146_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9147_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9148_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9149_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9150_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9151_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9152_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9153_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9154_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9155_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9156_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9157_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9158_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9159_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9160_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9161_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_9162_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_9163_real__root__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( inverse_inverse_real @ X ) )
      = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).

% real_root_inverse
thf(fact_9164_real__root__divide,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( divide_divide_real @ X @ Y ) )
      = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_divide
thf(fact_9165_real__root__commute,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ M @ ( root @ N @ X ) )
      = ( root @ N @ ( root @ M @ X ) ) ) ).

% real_root_commute
thf(fact_9166_real__root__mult__exp,axiom,
    ! [M: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M @ N ) @ X )
      = ( root @ M @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_9167_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_9168_real__root__minus,axiom,
    ! [N: nat,X: real] :
      ( ( root @ N @ ( uminus_uminus_real @ X ) )
      = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).

% real_root_minus
thf(fact_9169_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).

% real_root_pos_pos_le
thf(fact_9170_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9171_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9172_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K ) )
        = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).

% real_root_power
thf(fact_9173_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_9174_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_9175_real__root__strict__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9176_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9177_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9178_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_9179_real__root__strict__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9180_real__root__decreasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9181_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_9182_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9183_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_9184_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( power_power_real @ ( root @ N @ X ) @ N )
        = X ) ) ).

% odd_real_root_pow
thf(fact_9185_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y @ N )
          = X )
       => ( ( root @ N @ X )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9186_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( root @ N @ ( power_power_real @ X @ N ) )
        = X ) ) ).

% odd_real_root_power_cancel
thf(fact_9187_real__root__increasing,axiom,
    ! [N: nat,N4: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9188_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_9189_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_9190_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_9191_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9192_cot__gt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).

% cot_gt_zero
thf(fact_9193_tan__cot_H,axiom,
    ! [X: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
      = ( cot_real @ X ) ) ).

% tan_cot'
thf(fact_9194_arctan__def,axiom,
    ( arctan
    = ( ^ [Y3: real] :
          ( the_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 )
                = Y3 ) ) ) ) ) ).

% arctan_def
thf(fact_9195_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y3: real] :
          ( the_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 )
                = Y3 ) ) ) ) ) ).

% arcsin_def
thf(fact_9196_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9197_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = 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 @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = 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 @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
          & ( ( ( 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 @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9198_dvd__mult__sgn__iff,axiom,
    ! [L2: int,K: int,R: int] :
      ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_9199_dvd__sgn__mult__iff,axiom,
    ! [L2: int,R: int,K: int] :
      ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
      = ( ( dvd_dvd_int @ L2 @ K )
        | ( R = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_9200_mult__sgn__dvd__iff,axiom,
    ! [L2: int,R: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R ) ) @ K )
      = ( ( dvd_dvd_int @ L2 @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_9201_sgn__mult__dvd__iff,axiom,
    ! [R: int,L2: int,K: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L2 ) @ K )
      = ( ( dvd_dvd_int @ L2 @ K )
        & ( ( R = zero_zero_int )
         => ( K = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_9202_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9203_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N ) ) ).

% signed_take_bit_negative_iff
thf(fact_9204_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9205_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9206_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9207_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9208_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_and_int_iff
thf(fact_9209_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_9210_div__eq__sgn__abs,axiom,
    ! [K: int,L2: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( divide_divide_int @ K @ L2 )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_9211_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff'
thf(fact_9212_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_9213_ln__neg__is__const,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ln_ln_real @ X )
        = ( the_real
          @ ^ [X3: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9214_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I5: int] : ( if_int @ ( I5 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I5 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9215_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L2: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_9216_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9217_div__dvd__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( dvd_dvd_int @ L2 @ K )
     => ( ( divide_divide_int @ K @ L2 )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_9218_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
      = ( ( ( ord_less_nat @ N @ M )
          & ( bit_se1146084159140164899it_int @ K @ N ) )
        | ( ( ord_less_eq_nat @ M @ N )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9219_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9220_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N3 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9221_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K2: int,N2: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_int_def
thf(fact_9222_arccos__def,axiom,
    ( arccos
    = ( ^ [Y3: real] :
          ( the_real
          @ ^ [X3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X3 )
              & ( ord_less_eq_real @ X3 @ pi )
              & ( ( cos_real @ X3 )
                = Y3 ) ) ) ) ) ).

% arccos_def
thf(fact_9223_eucl__rel__int__remainderI,axiom,
    ! [R: int,L2: int,K: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R )
        = ( sgn_sgn_int @ L2 ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R ) )
         => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9224_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A32 )
     => ( ( ( A23 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R2: int,Q3: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q3 @ R2 ) )
               => ( ( ( sgn_sgn_int @ R2 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R2 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9225_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
          ( ? [K2: int] :
              ( ( A1 = K2 )
              & ( A22 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
          | ? [L: int,K2: int,Q5: int] :
              ( ( A1 = K2 )
              & ( A22 = L )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
              & ( L != zero_zero_int )
              & ( K2
                = ( times_times_int @ Q5 @ L ) ) )
          | ? [R5: int,L: int,K2: int,Q5: int] :
              ( ( A1 = K2 )
              & ( A22 = L )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
              & ( K2
                = ( plus_plus_int @ ( times_times_int @ Q5 @ L ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9226_div__noneq__sgn__abs,axiom,
    ! [L2: int,K: int] :
      ( ( L2 != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L2 ) )
       => ( ( divide_divide_int @ K @ L2 )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_9227_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] :
          ( plus_plus_int @ K2
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% set_bit_eq
thf(fact_9228_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9229_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [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 ) ) ) ) ).

% pi_half
thf(fact_9230_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [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 ) ) ) ) ) ).

% pi_def
thf(fact_9231_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9232_divide__int__unfold,axiom,
    ! [L2: int,K: int,N: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N = 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 @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N = 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 @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
          & ( ( ( 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 @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9233_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K2: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K2
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( 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 @ K2 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9234_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K2: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9235_powr__int,axiom,
    ! [X: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9236_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
      = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).

% zero_le_sgn_iff
thf(fact_9237_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
      = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9238_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9239_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9240_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9241_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9242_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_9243_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9244_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9245_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_9246_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_9247_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_9248_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9249_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9250_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9251_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_9252_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_9253_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9254_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9255_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9256_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9257_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_nat_iff
thf(fact_9258_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9259_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( N = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9260_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9261_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I5: num] : ( nat2 @ ( numeral_numeral_int @ I5 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9262_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_9263_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
          & ( P4 @ ( nat2 @ X3 ) ) ) ) ) ).

% ex_nat
thf(fact_9264_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P4 @ ( nat2 @ X3 ) ) ) ) ) ).

% all_nat
thf(fact_9265_eq__nat__nat__iff,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z7 ) )
          = ( Z = Z7 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9266_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9267_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9268_nat__mask__eq,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( bit_se2002935070580805687sk_nat @ N ) ) ).

% nat_mask_eq
thf(fact_9269_not__bit__Suc__0__numeral,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9270_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_9271_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_9272_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_9273_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_9274_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_9275_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9276_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_9277_int__minus,axiom,
    ! [N: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
      = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).

% int_minus
thf(fact_9278_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_9279_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% and_nat_def
thf(fact_9280_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9281_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9282_real__nat__ceiling__ge,axiom,
    ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_9283_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9284_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9285_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9286_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( cis @ ( arg @ Z ) )
        = ( sgn_sgn_complex @ Z ) ) ) ).

% cis_Arg
thf(fact_9287_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9288_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_9289_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_9290_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ( nat2 @ W )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_9291_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_9292_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N2: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N2 ) )
           => ( P @ N2 ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9293_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9294_nat__add__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9295_nat__mult__distrib,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9296_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9297_nat__diff__distrib,axiom,
    ! [Z7: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
     => ( ( ord_less_eq_int @ Z7 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9298_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9299_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_9300_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_9301_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_9302_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9303_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9304_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9305_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_9306_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_9307_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9308_mod__abs__eq__div__nat,axiom,
    ! [K: int,L2: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_9309_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
        = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9310_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9311_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9312_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9313_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_9314_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_9315_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z7: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9316_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_9317_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_9318_diff__nat__eq__if,axiom,
    ! [Z7: int,Z: int] :
      ( ( ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9319_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M3: nat,N2: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% bit_nat_def
thf(fact_9320_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9321_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_9322_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_9323_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
       => ( ( ord_less_eq_real @ X @ pi )
         => ( ( arg @ Z )
            = X ) ) ) ) ).

% cis_Arg_unique
thf(fact_9324_split__root,axiom,
    ! [P: real > $o,N: nat,X: real] :
      ( ( P @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y3: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
                = X )
             => ( P @ Y3 ) ) ) ) ) ).

% split_root
thf(fact_9325_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_9326_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_9327_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
         => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
            = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9328_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_9329_Arg__def,axiom,
    ( arg
    = ( ^ [Z3: complex] :
          ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A3: real] :
                ( ( ( sgn_sgn_complex @ Z3 )
                  = ( cis @ A3 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
                & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9330_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( ( K2
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( 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 @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9331_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_9332_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_9333_or__minus__numerals_I6_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9334_or__minus__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9335_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
        | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_or_int_iff
thf(fact_9336_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_9337_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).

% OR_lower
thf(fact_9338_plus__and__or,axiom,
    ! [X: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
      = ( plus_plus_int @ X @ Y ) ) ).

% plus_and_or
thf(fact_9339_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_9340_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9341_or__minus__numerals_I5_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9342_or__minus__numerals_I1_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9343_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9344_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_9345_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_9346_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_9347_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_9348_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9349_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9350_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9351_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9352_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9353_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9354_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9355_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9356_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_9357_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9358_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9359_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% or_nat_def
thf(fact_9360_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_9361_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9362_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9363_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa2 )
        = Y )
     => ( ( ( X = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X = one )
           => ! [M4: num] :
                ( ( Xa2
                  = ( bit0 @ M4 ) )
               => ( Y
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa2
                    = ( bit1 @ M4 ) )
                 => ( Y
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N3: num] :
                    ( X
                    = ( bit0 @ N3 ) )
               => ( ( Xa2 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N3: num] :
                    ( ( X
                      = ( bit0 @ N3 ) )
                   => ! [M4: num] :
                        ( ( Xa2
                          = ( bit0 @ M4 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X
                        = ( bit0 @ N3 ) )
                     => ! [M4: num] :
                          ( ( Xa2
                            = ( bit1 @ M4 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
                 => ( ( ? [N3: num] :
                          ( X
                          = ( bit1 @ N3 ) )
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N3: num] :
                          ( ( X
                            = ( bit1 @ N3 ) )
                         => ! [M4: num] :
                              ( ( Xa2
                                = ( bit0 @ M4 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
                     => ~ ! [N3: num] :
                            ( ( X
                              = ( bit1 @ N3 ) )
                           => ! [M4: num] :
                                ( ( Xa2
                                  = ( bit1 @ M4 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9364_sin__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9365_cos__integer__2pi,axiom,
    ! [N: real] :
      ( ( member_real @ N @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9366_or__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9367_Suc__0__or__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9368_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9369_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9370_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X @ Xa2 )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa2 ) )
       => ( ( ( X = one )
           => ( ( Xa2 = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X = one )
             => ! [M4: num] :
                  ( ( Xa2
                    = ( bit0 @ M4 ) )
                 => ( ( Y
                      = ( bit1 @ M4 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
           => ( ( ( X = one )
               => ! [M4: num] :
                    ( ( Xa2
                      = ( bit1 @ M4 ) )
                   => ( ( Y
                        = ( bit1 @ M4 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
             => ( ! [N3: num] :
                    ( ( X
                      = ( bit0 @ N3 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X
                        = ( bit0 @ N3 ) )
                     => ! [M4: num] :
                          ( ( Xa2
                            = ( bit0 @ M4 ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
                 => ( ! [N3: num] :
                        ( ( X
                          = ( bit0 @ N3 ) )
                       => ! [M4: num] :
                            ( ( Xa2
                              = ( bit1 @ M4 ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
                   => ( ! [N3: num] :
                          ( ( X
                            = ( bit1 @ N3 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
                     => ( ! [N3: num] :
                            ( ( X
                              = ( bit1 @ N3 ) )
                           => ! [M4: num] :
                                ( ( Xa2
                                  = ( bit0 @ M4 ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X
                                = ( bit1 @ N3 ) )
                             => ! [M4: num] :
                                  ( ( Xa2
                                    = ( bit1 @ M4 ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9371_xor__Suc__0__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9372_Suc__0__xor__eq,axiom,
    ! [N: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9373_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9374_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9375_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9376_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9377_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9378_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9379_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_9380_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9381_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9382_concat__bit__of__zero__1,axiom,
    ! [N: nat,L2: int] :
      ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_9383_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_9384_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_9385_push__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% push_bit_of_Suc_0
thf(fact_9386_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
      = ( ( bit_se1146084159140164899it_int @ K @ N )
       != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).

% bit_xor_int_iff
thf(fact_9387_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9388_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9389_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).

% XOR_lower
thf(fact_9390_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9391_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9392_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9393_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% xor_nat_def
thf(fact_9394_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9395_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_9396_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N2: nat,K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9397_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9398_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9399_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% push_bit_int_def
thf(fact_9400_push__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% push_bit_minus_one
thf(fact_9401_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9402_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9403_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] :
          ( if_int
          @ ( K2
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K2 )
            @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9404_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9405_finite__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9406_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_9407_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_9408_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9409_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9410_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9411_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).

% bit_not_int_iff
thf(fact_9412_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_9413_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9414_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9415_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9416_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9417_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K2: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9418_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9419_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_9420_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_9421_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9422_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9423_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9424_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9425_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_9426_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_9427_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_9428_and__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9429_or__not__numerals_I2_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9430_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_9431_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).

% bit_minus_int_iff
thf(fact_9432_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9433_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9434_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9435_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9436_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9437_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9438_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9439_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_9440_or__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9441_and__not__numerals_I3_J,axiom,
    ! [N: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9442_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_9443_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_9444_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9445_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9446_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9447_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9448_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9449_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9450_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9451_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( 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 @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9452_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9453_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_9454_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I4: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ N )
           => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
     => ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ N )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I5: nat] : ( times_times_nat @ ( A @ I5 ) @ ( B @ I5 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9455_finite__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9456_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9457_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_9458_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X2: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M8 @ M3 )
         => ! [N2: nat] :
              ( ( ord_less_eq_nat @ M8 @ N2 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X2 @ M3 ) @ ( X2 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9459_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_9460_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_9461_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9462_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_9463_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_9464_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_9465_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_9466_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
thf(fact_9467_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_9468_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_9469_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_9470_cos__Arg__i__mult__zero,axiom,
    ! [Y: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( ( re @ Y )
          = zero_zero_real )
       => ( ( cos_real @ ( arg @ Y ) )
          = zero_zero_real ) ) ) ).

% cos_Arg_i_mult_zero
thf(fact_9471_imaginary__unit_Osimps_I1_J,axiom,
    ( ( re @ imaginary_unit )
    = zero_zero_real ) ).

% imaginary_unit.simps(1)
thf(fact_9472_complex__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% complex_Re_le_cmod
thf(fact_9473_zero__complex_Osimps_I1_J,axiom,
    ( ( re @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(1)
thf(fact_9474_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_9475_plus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% plus_complex.simps(1)
thf(fact_9476_scaleR__complex_Osimps_I1_J,axiom,
    ! [R: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
      = ( times_times_real @ R @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_9477_minus__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).

% minus_complex.simps(1)
thf(fact_9478_abs__Re__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Re_le_cmod
thf(fact_9479_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_9480_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_9481_cos__n__Re__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% cos_n_Re_cis_pow_n
thf(fact_9482_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z3: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z3 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9483_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_9484_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_9485_complex__Im__fact,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri5044797733671781792omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_fact
thf(fact_9486_complex__Im__of__int,axiom,
    ! [Z: int] :
      ( ( im @ ( ring_17405671764205052669omplex @ Z ) )
      = zero_zero_real ) ).

% complex_Im_of_int
thf(fact_9487_complex__Im__of__nat,axiom,
    ! [N: nat] :
      ( ( im @ ( semiri8010041392384452111omplex @ N ) )
      = zero_zero_real ) ).

% complex_Im_of_nat
thf(fact_9488_Im__complex__of__real,axiom,
    ! [Z: real] :
      ( ( im @ ( real_V4546457046886955230omplex @ Z ) )
      = zero_zero_real ) ).

% Im_complex_of_real
thf(fact_9489_Im__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X @ N ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_9490_complex__Im__numeral,axiom,
    ! [V: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9491_Im__i__times,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( re @ Z ) ) ).

% Im_i_times
thf(fact_9492_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X @ N ) )
        = ( power_power_real @ ( re @ X ) @ N ) ) ) ).

% Re_power_real
thf(fact_9493_Re__i__times,axiom,
    ! [Z: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
      = ( uminus_uminus_real @ ( im @ Z ) ) ) ).

% Re_i_times
thf(fact_9494_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_9495_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
       => ( ( csqrt @ X )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_9496_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_9497_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_9498_zero__complex_Osimps_I2_J,axiom,
    ( ( im @ zero_zero_complex )
    = zero_zero_real ) ).

% zero_complex.simps(2)
thf(fact_9499_one__complex_Osimps_I2_J,axiom,
    ( ( im @ one_one_complex )
    = zero_zero_real ) ).

% one_complex.simps(2)
thf(fact_9500_plus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% plus_complex.simps(2)
thf(fact_9501_scaleR__complex_Osimps_I2_J,axiom,
    ! [R: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
      = ( times_times_real @ R @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_9502_minus__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).

% minus_complex.simps(2)
thf(fact_9503_complex__is__Int__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ ring_1_Ints_complex )
      = ( ( ( im @ Z )
          = zero_zero_real )
        & ? [I5: int] :
            ( ( re @ Z )
            = ( ring_1_of_int_real @ I5 ) ) ) ) ).

% complex_is_Int_iff
thf(fact_9504_abs__Im__le__cmod,axiom,
    ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).

% abs_Im_le_cmod
thf(fact_9505_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( times_times_complex @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).

% times_complex.simps(2)
thf(fact_9506_cmod__eq__Re,axiom,
    ! [Z: complex] :
      ( ( ( im @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( re @ Z ) ) ) ) ).

% cmod_eq_Re
thf(fact_9507_cmod__eq__Im,axiom,
    ! [Z: complex] :
      ( ( ( re @ Z )
        = zero_zero_real )
     => ( ( real_V1022390504157884413omplex @ Z )
        = ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_eq_Im
thf(fact_9508_Im__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( abs_abs_real @ ( re @ Z ) )
        = ( real_V1022390504157884413omplex @ Z ) )
     => ( ( im @ Z )
        = zero_zero_real ) ) ).

% Im_eq_0
thf(fact_9509_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( im @ X )
        = ( im @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_9510_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( ( re @ X )
        = ( re @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_9511_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( times_times_complex @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).

% times_complex.simps(1)
thf(fact_9512_plus__complex_Ocode,axiom,
    ( plus_plus_complex
    = ( ^ [X3: complex,Y3: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( plus_plus_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) ) ) ).

% plus_complex.code
thf(fact_9513_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R5: real,X3: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X3 ) ) @ ( times_times_real @ R5 @ ( im @ X3 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_9514_minus__complex_Ocode,axiom,
    ( minus_minus_complex
    = ( ^ [X3: complex,Y3: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( minus_minus_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) ) ) ).

% minus_complex.code
thf(fact_9515_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_9516_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_9517_sin__n__Im__cis__pow__n,axiom,
    ! [N: nat,A: real] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
      = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).

% sin_n_Im_cis_pow_n
thf(fact_9518_Re__exp,axiom,
    ! [Z: complex] :
      ( ( re @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).

% Re_exp
thf(fact_9519_Im__exp,axiom,
    ! [Z: complex] :
      ( ( im @ ( exp_complex @ Z ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).

% Im_exp
thf(fact_9520_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_9521_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X3: complex,Y3: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_9522_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z3: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z3 ) ) ) @ ( cis @ ( im @ Z3 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_9523_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_9524_Im__power2,axiom,
    ! [X: complex] :
      ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).

% Im_power2
thf(fact_9525_Re__power2,axiom,
    ! [X: complex] :
      ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_9526_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_9527_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9528_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_9529_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_9530_Re__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9531_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_9532_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_9533_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_9534_Im__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9535_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_9536_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_9537_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X3: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X3 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X3 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9538_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X3: complex,Y3: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9539_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_9540_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_9541_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_9542_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_9543_complex__is__Real__iff,axiom,
    ! [Z: complex] :
      ( ( member_complex @ Z @ real_V2521375963428798218omplex )
      = ( ( im @ Z )
        = zero_zero_real ) ) ).

% complex_is_Real_iff
thf(fact_9544_Complex__in__Reals,axiom,
    ! [X: real] : ( member_complex @ ( complex2 @ X @ zero_zero_real ) @ real_V2521375963428798218omplex ) ).

% Complex_in_Reals
thf(fact_9545_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_9546_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_9547_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( times_times_complex @ X @ Y ) )
      = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_mult
thf(fact_9548_complex__cnj__zero,axiom,
    ( ( cnj @ zero_zero_complex )
    = zero_zero_complex ) ).

% complex_cnj_zero
thf(fact_9549_complex__cnj__zero__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_complex ) ) ).

% complex_cnj_zero_iff
thf(fact_9550_complex__cnj__one__iff,axiom,
    ! [Z: complex] :
      ( ( ( cnj @ Z )
        = one_one_complex )
      = ( Z = one_one_complex ) ) ).

% complex_cnj_one_iff
thf(fact_9551_complex__cnj__one,axiom,
    ( ( cnj @ one_one_complex )
    = one_one_complex ) ).

% complex_cnj_one
thf(fact_9552_complex__cnj__power,axiom,
    ! [X: complex,N: nat] :
      ( ( cnj @ ( power_power_complex @ X @ N ) )
      = ( power_power_complex @ ( cnj @ X ) @ N ) ) ).

% complex_cnj_power
thf(fact_9553_complex__cnj__add,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( plus_plus_complex @ X @ Y ) )
      = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_add
thf(fact_9554_complex__cnj__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% complex_cnj_numeral
thf(fact_9555_complex__cnj__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
      = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_diff
thf(fact_9556_complex__cnj__neg__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_cnj_neg_numeral
thf(fact_9557_complex__In__mult__cnj__zero,axiom,
    ! [Z: complex] :
      ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_9558_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_9559_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_9560_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z3: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_9561_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_9562_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_9563_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_9564_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_9565_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_9566_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_9567_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_9568_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_9569_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_9570_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_9571_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_9572_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_9573_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9574_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_9575_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9576_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_9577_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_nat @ I5 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_9578_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9579_card__atLeastLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_atLeastLessThan
thf(fact_9580_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_eq_nat @ I5 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_9581_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_9582_card__atLeastLessThan__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_atLeastLessThan_int
thf(fact_9583_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_9584_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_9585_minus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
      = ( uminus1351360451143612070nteger @ L2 ) ) ).

% minus_integer_code(2)
thf(fact_9586_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_9587_times__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_9588_plus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
      = L2 ) ).

% plus_integer_code(2)
thf(fact_9589_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_9590_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9591_exhaustive__integer_H_Ocases,axiom,
    ! [X: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D3: code_integer,I4: code_integer] :
          ( X
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I4 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_9592_full__exhaustive__integer_H_Ocases,axiom,
    ! [X: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D3: code_integer,I4: code_integer] :
          ( X
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D3 @ I4 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_9593_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9594_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9595_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_9596_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_9597_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9598_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M5 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9599_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K2: nat] :
                  ( ( member_nat @ ( suc @ K2 ) @ M5 )
                  & ( ord_less_nat @ K2 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M5 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9600_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ ( suc @ K2 ) @ M5 )
                & ( ord_less_nat @ K2 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K2: nat] :
                ( ( member_nat @ K2 @ M5 )
                & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9601_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9602_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_9603_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_9604_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_9605_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9606_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9607_max__Suc2,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9608_max__Suc1,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9609_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9610_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X3: nat] : X3
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_9611_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z3: complex] :
                  ( ( power_power_complex @ Z3 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9612_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9613_diff__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K2: nat] : K2
        @ ( minus_minus_nat @ M @ N ) ) ) ).

% diff_Suc
thf(fact_9614_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K2: int] :
          ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
          @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9615_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).

% abs_integer_code
thf(fact_9616_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_9617_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_9618_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9619_modulo__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa2 @ X ) ) ) ).

% modulo_integer.abs_eq
thf(fact_9620_plus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( plus_plus_int @ Xa2 @ X ) ) ) ).

% plus_integer.abs_eq
thf(fact_9621_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_9622_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_9623_less__eq__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( ord_less_eq_int @ Xa2 @ X ) ) ).

% less_eq_integer.abs_eq
thf(fact_9624_minus__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).

% minus_integer.abs_eq
thf(fact_9625_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_9626_integer__of__num_I3_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).

% integer_of_num(3)
thf(fact_9627_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_9628_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9629_integer__of__num_I2_J,axiom,
    ! [N: num] :
      ( ( code_integer_of_num @ ( bit0 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).

% integer_of_num(2)
thf(fact_9630_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9631_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K2: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
          @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9632_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9633_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K2: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ 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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9634_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_9635_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_9636_plus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa2 ) )
      = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% plus_integer.rep_eq
thf(fact_9637_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_9638_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_9639_minus__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
      = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% minus_integer.rep_eq
thf(fact_9640_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X @ Xa2 ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% modulo_integer.rep_eq
thf(fact_9641_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K2: code_integer,L: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9642_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9643_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L ) @ ( modulo364778990260209775nteger @ K2 @ L ) ) ) ) ).

% divmod_integer_def
thf(fact_9644_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K2: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9645_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K2: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ 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 @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9646_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_9647_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_9648_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_9649_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_9650_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_9651_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_9652_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_9653_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9654_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9655_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S7 ) ) )
                @ ( code_divmod_abs @ K2 @ L ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S7 ) ) )
                    @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9656_binomial__def,axiom,
    ( binomial
    = ( ^ [N2: nat,K2: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K2 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9657_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9658_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X3: real] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9659_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X3: rat] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9660_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_9661_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_9662_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9663_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9664_drop__bit__minus__one,axiom,
    ! [N: nat] :
      ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9665_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( modulo_modulo_nat @ M @ N ) ) ).

% snd_divmod_nat
thf(fact_9666_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9667_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_9668_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9669_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X3: rat,Y3: rat] :
          ( ( ord_less_rat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9670_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_rat_def
thf(fact_9671_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9672_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R
                 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9673_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9674_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9675_rat__inverse__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B2 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_inverse_code
thf(fact_9676_normalize__negative,axiom,
    ! [Q2: int,P5: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P5 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9677_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9678_snd__divmod__integer,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L2 ) )
      = ( modulo364778990260209775nteger @ K @ L2 ) ) ).

% snd_divmod_integer
thf(fact_9679_snd__divmod__abs,axiom,
    ! [K: code_integer,L2: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L2 ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).

% snd_divmod_abs
thf(fact_9680_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_9681_normalize__denom__zero,axiom,
    ! [P5: int] :
      ( ( normalize @ ( product_Pair_int_int @ P5 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9682_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9683_drop__bit__of__Suc__0,axiom,
    ! [N: nat] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9684_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9685_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_9686_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_9687_quotient__of__denom__pos_H,axiom,
    ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9688_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q5: rat,R5: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).

% divide_rat_def
thf(fact_9689_diff__rat__def,axiom,
    ( minus_minus_rat
    = ( ^ [Q5: rat,R5: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).

% diff_rat_def
thf(fact_9690_rat__divide__code,axiom,
    ! [P5: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P5 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_divide_code
thf(fact_9691_rat__times__code,axiom,
    ! [P5: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P5 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B2 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_times_code
thf(fact_9692_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9693_quotient__of__div,axiom,
    ! [R: rat,N: int,D: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ N @ D ) )
     => ( R
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9694_rat__plus__code,axiom,
    ! [P5: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P5 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_plus_code
thf(fact_9695_rat__minus__code,axiom,
    ! [P5: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P5 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_minus_code
thf(fact_9696_quotient__of__denom__pos,axiom,
    ! [R: rat,P5: int,Q2: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ P5 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9697_rat__uminus__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_uminus_code
thf(fact_9698_rat__abs__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P5 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
        @ ( quotient_of @ P5 ) ) ) ).

% rat_abs_code
thf(fact_9699_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P5: int,Q2: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P5 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9700_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P5: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P5 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
         => ( ( times_times_int @ P5 @ S )
            = ( times_times_int @ R @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9701_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9702_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P2: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P2 ) ) ) ) ).

% rat_less_code
thf(fact_9703_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P2: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P2 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9704_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9705_minus__one__mod__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9706_one__mod__minus__numeral,axiom,
    ! [N: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9707_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9708_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9709_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9710_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9711_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9712_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_9713_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X3: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9714_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9715_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9716_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9717_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9718_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_9719_rat__sgn__code,axiom,
    ! [P5: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P5 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P5 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9720_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_9721_Frct__code__post_I6_J,axiom,
    ! [K: num,L2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).

% Frct_code_post(6)
thf(fact_9722_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9723_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9724_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9725_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
      = ( divide_divide_nat @ M @ N ) ) ).

% fst_divmod_nat
thf(fact_9726_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_9727_normalize__def,axiom,
    ( normalize
    = ( ^ [P2: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P2 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P2 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P2 ) @ ( product_snd_int_int @ P2 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P2 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P2 ) @ ( product_snd_int_int @ P2 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P2 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P2 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P2 ) @ ( product_snd_int_int @ P2 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P2 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P2 ) @ ( product_snd_int_int @ P2 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9728_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9729_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
      = ( ( M != zero_zero_int )
        | ( N != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9730_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9731_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] :
      ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9732_gcd__0__left__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_left_int
thf(fact_9733_gcd__0__int,axiom,
    ! [X: int] :
      ( ( gcd_gcd_int @ X @ zero_zero_int )
      = ( abs_abs_int @ X ) ) ).

% gcd_0_int
thf(fact_9734_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9735_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V2 @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9736_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X3: int,Y3: int] : ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).

% gcd_red_int
thf(fact_9737_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9738_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).

% gcd_le1_int
thf(fact_9739_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).

% gcd_le2_int
thf(fact_9740_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9741_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_9742_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ Y )
     => ( ( gcd_gcd_int @ X @ Y )
        = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_non_0_int
thf(fact_9743_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K2: int,L: int] : ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( gcd_gcd_int @ L @ ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_9744_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K3: nat] :
          ( ( ord_less_nat @ N @ K3 )
         => ( P @ K3 ) )
     => ( ! [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K3 @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K3 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9745_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R2: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N7: nat] :
              ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R2 @ N7 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_9746_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K2: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K2 )
                  = ( sgn_sgn_Code_integer @ L ) )
                @ ( code_divmod_abs @ K2 @ L )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S7 ) ) )
                  @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9747_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9748_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9749_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9750_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9751_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9752_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9753_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9754_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9755_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
      = ( ( M != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9756_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9757_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9758_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
        = ( gcd_gcd_nat @ M @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9759_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X3: nat,Y3: nat] : ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ).

% gcd_red_nat
thf(fact_9760_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9761_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X3: nat,Y3: nat] : ( if_nat @ ( Y3 = zero_zero_nat ) @ X3 @ ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9762_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9763_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_9764_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_9765_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X6: nat,Y5: nat] :
          ( ( times_times_nat @ A @ X6 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9766_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X6: nat,Y5: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y5 ) @ ( times_times_nat @ A @ X6 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X6 ) @ ( times_times_nat @ B @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y5 ) @ ( times_times_nat @ B @ X6 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X6 ) @ ( times_times_nat @ A @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9767_gcd__code__integer,axiom,
    ( gcd_gcd_Code_integer
    = ( ^ [K2: code_integer,L: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L = zero_z3403309356797280102nteger ) @ K2 @ ( gcd_gcd_Code_integer @ L @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ) ) ).

% gcd_code_integer
thf(fact_9768_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9769_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9770_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_9771_Code__Target__Int_Onegative__def,axiom,
    ( code_Target_negative
    = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).

% Code_Target_Int.negative_def
thf(fact_9772_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_9773_finite__greaterThanLessThan__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L2 @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9774_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_9775_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9776_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9777_finite__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9778_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_9779_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
      = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9780_tanh__real__bounds,axiom,
    ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9781_sub__BitM__One__eq,axiom,
    ! [N: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9782_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9783_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X3 )
    @ ^ [X3: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X3 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9784_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M3: nat,N2: nat] :
        ( ( dvd_dvd_nat @ M3 @ N2 )
        & ( M3 != N2 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9785_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9786_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( gcd_Gcd_nat @ M5 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9787_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9788_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X6: nat,Y5: nat] :
          ( Z
         != ( abs_Integ @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) ) ).

% eq_Abs_Integ
thf(fact_9789_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( ( member_nat @ one_one_nat @ N4 )
     => ( ( gcd_Gcd_nat @ N4 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9790_nat_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).

% nat.abs_eq
thf(fact_9791_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9792_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9793_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9794_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9795_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X ) ) ).

% less_int.abs_eq
thf(fact_9796_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9797_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9798_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X3: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9799_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_9800_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9801_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z3: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
          @ ( rep_Integ @ X3 )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9802_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X3: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X3 ) ) ) ) ).

% nat.rep_eq
thf(fact_9803_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).

% prod_encode_def
thf(fact_9804_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_9805_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_9806_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9807_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X3: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X3 ) ) ) ) ).

% uminus_int_def
thf(fact_9808_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X3 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X3 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9809_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9810_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X3: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X3 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9811_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = ( inc @ ( num_of_nat @ N ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( suc @ N ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9812_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9813_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9814_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
        = N ) ) ).

% numeral_num_of_nat
thf(fact_9815_num__of__nat__One,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ one_one_nat )
     => ( ( num_of_nat @ N )
        = one ) ) ).

% num_of_nat_One
thf(fact_9816_num__of__nat__double,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
        = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).

% num_of_nat_double
thf(fact_9817_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9818_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9819_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9820_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9821_finite__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9822_card__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
      = ( minus_minus_nat @ U @ L2 ) ) ).

% card_greaterThanAtMost
thf(fact_9823_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L2: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
      = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9824_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9825_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9826_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X3: num] : ( times_times_num @ X3 @ X3 ) ) ) ).

% sqr_conv_mult
thf(fact_9827_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9828_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9829_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_9830_finite__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L2 @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9831_mult__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).

% mult_rat
thf(fact_9832_divide__rat,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).

% divide_rat
thf(fact_9833_card__greaterThanAtMost__int,axiom,
    ! [L2: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_9834_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9835_add__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_9836_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9837_diff__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9838_sgn__rat,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).

% sgn_rat
thf(fact_9839_Fract__of__int__eq,axiom,
    ! [K: int] :
      ( ( fract @ K @ one_one_int )
      = ( ring_1_of_int_rat @ K ) ) ).

% Fract_of_int_eq
thf(fact_9840_quotient__of__eq,axiom,
    ! [A: int,B: int,P5: int,Q2: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P5 @ Q2 ) )
     => ( ( fract @ P5 @ Q2 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_9841_One__rat__def,axiom,
    ( one_one_rat
    = ( fract @ one_one_int @ one_one_int ) ) ).

% One_rat_def
thf(fact_9842_rat__number__collapse_I1_J,axiom,
    ! [K: int] :
      ( ( fract @ zero_zero_int @ K )
      = zero_zero_rat ) ).

% rat_number_collapse(1)
thf(fact_9843_rat__number__collapse_I6_J,axiom,
    ! [K: int] :
      ( ( fract @ K @ zero_zero_int )
      = zero_zero_rat ) ).

% rat_number_collapse(6)
thf(fact_9844_mult__rat__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_9845_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9846_Fract__of__nat__eq,axiom,
    ! [K: nat] :
      ( ( fract @ ( semiri1314217659103216013at_int @ K ) @ one_one_int )
      = ( semiri681578069525770553at_rat @ K ) ) ).

% Fract_of_nat_eq
thf(fact_9847_eq__rat_I2_J,axiom,
    ! [A: int] :
      ( ( fract @ A @ zero_zero_int )
      = ( fract @ zero_zero_int @ one_one_int ) ) ).

% eq_rat(2)
thf(fact_9848_eq__rat_I3_J,axiom,
    ! [A: int,C: int] :
      ( ( fract @ zero_zero_int @ A )
      = ( fract @ zero_zero_int @ C ) ) ).

% eq_rat(3)
thf(fact_9849_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q2: rat] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ zero_zero_int @ B5 )
         => ( P @ ( fract @ A5 @ B5 ) ) )
     => ( P @ Q2 ) ) ).

% Rat_induct_pos
thf(fact_9850_normalize__eq,axiom,
    ! [A: int,B: int,P5: int,Q2: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P5 @ Q2 ) )
     => ( ( fract @ P5 @ Q2 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_9851_Zero__rat__def,axiom,
    ( zero_zero_rat
    = ( fract @ zero_zero_int @ one_one_int ) ) ).

% Zero_rat_def
thf(fact_9852_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_9853_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_9854_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_9855_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_9856_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_9857_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9858_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_9859_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_9860_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_9861_Fract__add__one,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
        = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).

% Fract_add_one
thf(fact_9862_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9863_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_9864_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_9865_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_9866_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_9867_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_9868_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I5: nat] : ( minus_minus_nat @ I5 @ C )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9869_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9870_bij__betw__Suc,axiom,
    ! [M5: set_nat,N4: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M5 @ N4 )
      = ( ( image_nat_nat @ suc @ M5 )
        = N4 ) ) ).

% bij_betw_Suc
thf(fact_9871_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_9872_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_9873_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9874_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9875_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X3: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X3 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ) ).

% less_int_def
thf(fact_9876_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9877_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9878_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9879_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9880_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9881_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9882_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9883_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9884_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9885_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X2: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X2 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9886_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9887_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S6: set_int] :
        ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9888_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9889_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9890_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
              @ ^ [I5: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I5 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9891_image__add__int__atLeastLessThan,axiom,
    ! [L2: int,U: int] :
      ( ( image_int_int
        @ ^ [X3: int] : ( plus_plus_int @ X3 @ L2 )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
      = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9892_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9893_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9894_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_9895_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9896_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9897_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_9898_root__def,axiom,
    ( root
    = ( ^ [N2: nat,X3: real] :
          ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N2 ) )
            @ X3 ) ) ) ) ).

% root_def
thf(fact_9899_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_9900_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_9901_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_9902_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_9903_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_9904_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_9905_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_9906_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_9907_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_9908_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_9909_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_9910_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_9911_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X6: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X6 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X6 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9912_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I5: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I5 ) @ Js @ ( upto_aux @ I5 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9913_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_9914_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9915_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_9916_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_9917_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9918_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X6: nat,Xs3: list_nat] :
                ( ( X
                  = ( cons_nat @ X6 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X6 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X6 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9919_upto_Opelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9920_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9921_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9922_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9923_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_9924_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_9925_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_9926_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9927_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9928_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = ( 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 @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9929_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( 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 @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9930_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I5: int,J3: int] : ( append_int @ ( upto @ I5 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9931_upto__code,axiom,
    ( upto
    = ( ^ [I5: int,J3: int] : ( upto_aux @ I5 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9932_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_9933_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ I5 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9934_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_9935_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_9936_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ I5 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9937_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9938_upto_Oelims,axiom,
    ! [X: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa2 )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa2 )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9939_upto_Osimps,axiom,
    ( upto
    = ( ^ [I5: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I5 @ J3 ) @ ( cons_int @ I5 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9940_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_9941_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_9942_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I5: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I5 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9943_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_9944_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_9945_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9946_DERIV__local__const,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D )
             => ( ( F @ X )
                = ( F @ Y5 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9947_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ 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 @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9948_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ 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 @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9949_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X6: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X6 @ 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_9950_DERIV__const__ratio__const2,axiom,
    ! [A: real,B: real,F: real > real,K: real] :
      ( ( A != B )
     => ( ! [X6: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X6 @ 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_9951_DERIV__isconst__all,axiom,
    ! [F: real > real,X: real,Y: real] :
      ( ! [X6: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
     => ( ( F @ X )
        = ( F @ Y ) ) ) ).

% DERIV_isconst_all
thf(fact_9952_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ 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 @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9953_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ 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 @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9954_DERIV__ln,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9955_DERIV__isconst3,axiom,
    ! [A: real,B: real,X: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X6: real] :
                ( ( member_real @ X6 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) )
           => ( ( F @ X )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9956_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( 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 @ X @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9957_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( 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 @ X @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9958_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( 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 @ X @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9959_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
     => ( ( 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 @ X @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9960_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_eq_real @ A @ X6 )
           => ( ( ord_less_eq_real @ X6 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
       => ? [Z5: real] :
            ( ( ord_less_real @ A @ Z5 )
            & ( ord_less_real @ Z5 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z5 ) ) ) ) ) ) ).

% MVT2
thf(fact_9961_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_eq_real @ A @ X6 )
           => ( ( ord_less_eq_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_real @ Y2 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9962_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_eq_real @ A @ X6 )
           => ( ( ord_less_eq_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y2 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9963_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X6 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9964_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_eq_real @ A @ X6 )
           => ( ( ord_less_eq_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y2 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9965_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_eq_real @ A @ X6 )
           => ( ( ord_less_eq_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9966_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X6: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X6 @ 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_9967_DERIV__local__min,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y5 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9968_DERIV__local__max,axiom,
    ! [F: real > real,L2: real,X: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9969_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9970_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X3: real] : ( power_power_real @ X3 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S ) ) ).

% DERIV_pow
thf(fact_9971_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X3: real] : ( power_power_real @ ( G @ X3 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9972_has__real__derivative__powr,axiom,
    ! [Z: real,R: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z3: real] : ( powr_real @ Z3 @ 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_9973_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9974_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X: real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ R )
          @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9975_DERIV__powr,axiom,
    ! [G: real > real,M: real,X: real,F: real > real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X3: real] : ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9976_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9977_DERIV__series_H,axiom,
    ! [F: real > nat > real,F5: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N3: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X3: real] : ( F @ X3 @ N3 )
          @ ( F5 @ X0 @ N3 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X6 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F5 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N3: nat,X6: real,Y5: real] :
                    ( ( member_real @ X6 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y5 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X6 @ N3 ) @ ( F @ Y5 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X6 @ Y5 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X3: real] : ( suminf_real @ ( F @ X3 ) )
                  @ ( suminf_real @ ( F5 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9978_DERIV__arctan,axiom,
    ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9979_arsinh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9980_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9981_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9982_artanh__real__has__field__derivative,axiom,
    ! [X: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9983_DERIV__power__series_H,axiom,
    ! [R4: real,F: nat > real,X0: real] :
      ( ! [X6: real] :
          ( ( member_real @ X6 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X6 @ N2 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
       => ( ( ord_less_real @ zero_zero_real @ R4 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X3: real] :
                ( suminf_real
                @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) )
            @ ( suminf_real
              @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9984_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9985_DERIV__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9986_DERIV__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9987_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X6: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9988_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X6: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9989_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9990_Maclaurin__minus,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ H2 @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H2 @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9991_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9992_Maclaurin,axiom,
    ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T6: real] :
                ( ( ( ord_less_nat @ M4 @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9993_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M4: nat,X6: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9994_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T6: real] :
            ( ( ( ord_less_nat @ M4 @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9995_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9996_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9997_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T6: real] :
              ( ( ( ord_less_nat @ M4 @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9998_Maclaurin__lemma2,axiom,
    ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
      ( ! [M4: nat,T6: real] :
          ( ( ( ord_less_nat @ M4 @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K ) )
       => ! [M2: nat,T7: real] :
            ( ( ( ord_less_nat @ M2 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P2 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P2 ) ) @ ( power_power_real @ U2 @ P2 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
                    @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P2 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P2 ) ) @ ( power_power_real @ T7 @ P2 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9999_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_10000_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_10001_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q5: 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 @ Q5 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_10002_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_10003_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_10004_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_10005_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_10006_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_10007_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
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_10008_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_10009_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
            @ ( bit_take_bit_num @ N2 @ M ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_10010_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_10011_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_10012_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_10013_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_10014_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_10015_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_10016_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_10017_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_10018_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_10019_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_10020_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_10021_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_10022_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_10023_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N9: num] : ( some_num @ ( bit1 @ N9 ) )
        @ ( bit_and_not_num @ M @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_10024_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( ( bit_and_not_num @ M @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_10025_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_10026_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_10027_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N2: nat,M3: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X3: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P2: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P2 ) )
                  @ ^ [P2: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P2 ) ) )
                  @ X3 )
              @ A3 )
          @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_10028_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ( ord_less_eq_real @ A @ X6 )
              & ( ord_less_eq_real @ X6 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: 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 ) @ M9 ) ) )
            & ! [Y2: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y2 )
                  & ( ord_less_eq_real @ Y2 @ M9 ) )
               => ? [X6: real] :
                    ( ( ord_less_eq_real @ A @ X6 )
                    & ( ord_less_eq_real @ X6 @ B )
                    & ( ( F @ X6 )
                      = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_10029_isCont__real__sqrt,axiom,
    ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).

% isCont_real_sqrt
thf(fact_10030_isCont__real__root,axiom,
    ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).

% isCont_real_root
thf(fact_10031_isCont__inverse__function2,axiom,
    ! [A: real,X: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X )
     => ( ( ord_less_real @ X @ B )
       => ( ! [Z5: real] :
              ( ( ord_less_eq_real @ A @ Z5 )
             => ( ( ord_less_eq_real @ Z5 @ B )
               => ( ( G @ ( F @ Z5 ) )
                  = Z5 ) ) )
         => ( ! [Z5: real] :
                ( ( ord_less_eq_real @ A @ Z5 )
               => ( ( ord_less_eq_real @ Z5 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_10032_isCont__ln,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ln_ln_real ) ) ).

% isCont_ln
thf(fact_10033_isCont__arcosh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_10034_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X )
         => ( ( ord_less_real @ X @ B )
           => ( ! [Y5: real] :
                  ( ( ord_less_real @ A @ Y5 )
                 => ( ( ord_less_real @ Y5 @ B )
                   => ( ( F @ ( G @ Y5 ) )
                      = Y5 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10035_isCont__arccos,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_10036_isCont__arcsin,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_10037_LIM__less__bound,axiom,
    ! [B: real,X: real,F: real > real] :
      ( ( ord_less_real @ B @ X )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ ( set_or1633881224788618240n_real @ B @ X ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X6 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10038_isCont__artanh,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_10039_isCont__inverse__function,axiom,
    ! [D: real,X: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z5: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
           => ( ( G @ ( F @ Z5 ) )
              = Z5 ) )
       => ( ! [Z5: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10040_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z5: real] :
            ( ( ord_less_eq_real @ A @ Z5 )
           => ( ( ord_less_eq_real @ Z5 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z5: real] :
              ( ( ord_less_eq_real @ A @ Z5 )
             => ( ( ord_less_eq_real @ Z5 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z5: real] :
                ( ( ord_less_real @ A @ Z5 )
               => ( ( ord_less_real @ Z5 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
           => ( ! [Z5: real] :
                  ( ( ord_less_real @ A @ Z5 )
                 => ( ( ord_less_real @ Z5 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10041_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ( gcd_Gcd_nat @ M5 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M3: nat] :
                      ( collect_nat
                      @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M3 ) )
                  @ M5 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_10042_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_10043_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
               => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_10044_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
               => ( ( F @ X4 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_10045_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R2 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_10046_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10047_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X2: set_nat] : ( if_nat @ ( X2 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X2 ) ) ) ) ).

% Sup_nat_def
thf(fact_10048_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N2: nat] :
          ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_10049_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_10050_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( cos_real @ X3 ) @ ( sin_real @ X3 ) )
    @ ( 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_10051_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 )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10052_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 ) )
         => ! [N7: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10053_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N ) ) )
        = ( abs_abs_int @ N ) ) ) ).

% Max_divisors_self_int
thf(fact_10054_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X3: nat] : ( times_times_nat @ X3 @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_10055_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_10056_monoseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B3 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_10057_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_10058_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D2: int] :
                ( ( dvd_dvd_int @ D2 @ M )
                & ( dvd_dvd_int @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_10059_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_10060_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R2: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_real @ R2 @ ( X8 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( X8 @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10061_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10062_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( root @ N2 @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_10063_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N2: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_10064_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10065_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L2: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L2 )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N7: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N7 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_10066_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10067_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10068_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_10069_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_10070_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10071_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10072_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10073_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10074_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
          @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10075_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( summable_real
            @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).

% summable
thf(fact_10076_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K3: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10077_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K3: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10078_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
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10079_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10080_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10081_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10082_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N7: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N7: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N2: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10083_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
          @ ^ [N2: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10084_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10085_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N2: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I5: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I5 ) @ ( A @ I5 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10086_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F6: real > real] :
        ? [C3: real] :
          ( F6
          = ( ^ [X3: real] : ( times_times_real @ X3 @ C3 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_10087_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y3: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y3 ) ) @ ( divide_divide_real @ one_one_real @ Y3 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10088_dist__real__def,axiom,
    ( real_V975177566351809787t_real
    = ( ^ [X3: real,Y3: real] : ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ).

% dist_real_def
thf(fact_10089_dist__complex__def,axiom,
    ( real_V3694042436643373181omplex
    = ( ^ [X3: complex,Y3: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ) ).

% dist_complex_def
thf(fact_10090_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_10091_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10092_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_10093_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_10094_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_10095_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_10096_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10097_exp__at__bot,axiom,
    filterlim_real_real @ exp_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_bot_real ).

% exp_at_bot
thf(fact_10098_filterlim__inverse__at__bot__neg,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_bot_neg
thf(fact_10099_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_10100_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_10101_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_10102_ln__at__0,axiom,
    filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% ln_at_0
thf(fact_10103_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_10104_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X6: real] :
          ( ( ord_less_eq_real @ X6 @ B )
         => ? [Y2: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y2 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10105_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10106_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_10107_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X3: real] : ( power_power_real @ ( F @ X3 ) @ N )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10108_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_10109_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_10110_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10111_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10112_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_10113_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X3: real] : ( divide_divide_real @ ( ln_ln_real @ X3 ) @ X3 )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10114_filterlim__inverse__at__top__right,axiom,
    filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).

% filterlim_inverse_at_top_right
thf(fact_10115_filterlim__inverse__at__right__top,axiom,
    filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).

% filterlim_inverse_at_right_top
thf(fact_10116_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X3: real] : ( divide_divide_real @ ( power_power_real @ X3 @ K ) @ ( exp_real @ X3 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10117_tendsto__exp__limit__at__top,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y3: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y3 ) ) @ Y3 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10118_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_10119_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X6: real] :
          ( ( ord_less_eq_real @ B @ X6 )
         => ? [Y2: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
              & ( ord_less_real @ Y2 @ 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_10120_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_10121_lhopital__left__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10122_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10123_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X6: nat] :
          ( ( ord_less_eq_nat @ C @ X6 )
         => ( P @ X6 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10124_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N2: nat] :
            ( ( ord_less_eq_nat @ N6 @ N2 )
           => ( P @ N2 ) ) ) ) ).

% eventually_sequentially
thf(fact_10125_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F3 ) ) ) ).

% le_sequentially
thf(fact_10126_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I5: nat] : ( P @ ( plus_plus_nat @ I5 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10127_eventually__at__right__to__0,axiom,
    ! [P: real > $o,A: real] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( plus_plus_real @ X3 @ A ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_right_to_0
thf(fact_10128_eventually__at__top__to__right,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ at_top_real )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% eventually_at_top_to_right
thf(fact_10129_eventually__at__right__to__top,axiom,
    ! [P: real > $o] :
      ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
      = ( eventually_real
        @ ^ [X3: real] : ( P @ ( inverse_inverse_real @ X3 ) )
        @ at_top_real ) ) ).

% eventually_at_right_to_top
thf(fact_10130_lhopital,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10131_lhopital__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10132_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10133_lhopital__right,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right
thf(fact_10134_lhopital__right__0,axiom,
    ! [F0: real > real,G0: real > real,G2: real > real,F5: real > real,F3: 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
            @ ^ [X3: real] :
                ( ( G0 @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F0 @ X3 ) @ ( G0 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).

% lhopital_right_0
thf(fact_10135_lhopital__left,axiom,
    ! [F: real > real,X: real,G: real > real,G2: real > real,F5: real > real,F3: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] :
                ( ( G @ X3 )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] :
                  ( ( G2 @ X3 )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
           => ( ( eventually_real
                @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
             => ( ( eventually_real
                  @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
                 => ( filterlim_real_real
                    @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                    @ F3
                    @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10136_lhopital__right__at__top,axiom,
    ! [G: real > real,X: real,G2: real > real,F: real > real,F5: real > real,Y: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ Y )
                @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).

% lhopital_right_at_top
thf(fact_10137_lhopital__right__0__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F5: real > real,X: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
     => ( ( eventually_real
          @ ^ [X3: real] :
              ( ( G2 @ X3 )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
       => ( ( eventually_real
            @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
         => ( ( eventually_real
              @ ^ [X3: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
           => ( ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F5 @ X3 ) @ ( G2 @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
             => ( filterlim_real_real
                @ ^ [X3: real] : ( divide_divide_real @ ( F @ X3 ) @ ( G @ X3 ) )
                @ ( topolo2815343760600316023s_real @ X )
                @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).

% lhopital_right_0_at_top
thf(fact_10138_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_10139_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_10140_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_10141_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_10142_Bseq__realpow,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10143_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X6: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X6 @ ( 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
                  @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                  @ ( 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 ) )
                        & ! [I5: nat] :
                            ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X3: nat] :
                                ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                 => ( ( ord_less_nat @ Mi3 @ X3 )
                                    & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_10144_decseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10145_decseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( 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_10146_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 )
        & ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X3 @ ( 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
          @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X2 )
            & ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
          @ ( 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 ) )
                & ! [I5: nat] :
                    ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I5 ) @ X2 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I5 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X3: nat] :
                        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X3 )
                         => ( ( ord_less_nat @ Mi3 @ X3 )
                            & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_10147_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [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
                        @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                        @ ( 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_10148_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [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
                    @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                      & ! [X3: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                    @ ( 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 ) )
                          & ! [I5: nat] :
                              ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X3: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X3: nat] :
                                  ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                   => ( ( ord_less_nat @ Mi3 @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_10149_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X6: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X6 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X6 @ ( 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
                      @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                        & ! [X3: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                      @ ( 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 ) )
                            & ! [I5: nat] :
                                ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X3: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X3: nat] :
                                    ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                     => ( ( ord_less_nat @ Mi3 @ X3 )
                                        & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_10150_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [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
                        @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                        @ ( 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_10151_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X2: set_int] :
          ( the_int
          @ ^ [X3: int] :
              ( ( member_int @ X3 @ X2 )
              & ! [Y3: int] :
                  ( ( member_int @ Y3 @ X2 )
                 => ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_10152_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [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
                        @ ( ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X2 )
                          & ! [X3: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                        @ ( 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 ) )
                              & ! [I5: nat] :
                                  ( ( ord_less_nat @ I5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I5 ) @ X2 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I5 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X3: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X2: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X2 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X3: nat] :
                                      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X3 )
                                       => ( ( ord_less_nat @ Mi3 @ X3 )
                                          & ( ord_less_eq_nat @ X3 @ 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_10153_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ( ord_less_eq_real @ A @ X6 )
              & ( ord_less_eq_real @ X6 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) @ F ) )
       => ( ! [X6: real] :
              ( ( ( ord_less_real @ A @ X6 )
                & ( ord_less_real @ X6 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) )
         => ( ! [X6: real] :
                ( ( ( ord_less_eq_real @ A @ X6 )
                  & ( ord_less_eq_real @ X6 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) @ G ) )
           => ( ! [X6: real] :
                  ( ( ( ord_less_real @ A @ X6 )
                    & ( ord_less_real @ X6 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_10154_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X6: real] :
              ( ( ord_less_real @ A @ X6 )
             => ( ( ord_less_real @ X6 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z5: real] :
              ( ( ord_less_real @ A @ Z5 )
              & ( ord_less_real @ Z5 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10155_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X6 ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X3: real] : ( arcosh_real @ ( F @ X3 ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10156_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ? [C2: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
            & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10157_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_10158_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_10159_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_10160_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X6: real] :
            ( ( member_real @ X6 @ A2 )
           => ( member_real @ ( F @ X6 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X3: real] : ( artanh_real @ ( F @ X3 ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10161_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X6: real] :
                ( ( ord_less_real @ A @ X6 )
               => ( ( ord_less_real @ X6 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
           => ? [Z5: real] :
                ( ( ord_less_real @ A @ Z5 )
                & ( ord_less_real @ Z5 @ B )
                & ( ( F5 @ Z5 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10162_mvt,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X6: real] :
              ( ( ord_less_real @ A @ X6 )
             => ( ( ord_less_real @ X6 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F5 @ X6 ) @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F5 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10163_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X6: real] :
              ( ( ord_less_real @ A @ X6 )
             => ( ( ord_less_real @ X6 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10164_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_real @ A @ X6 )
           => ( ( ord_less_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_real @ Y2 @ 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_10165_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X6: real] :
            ( ( ord_less_real @ A @ X6 )
           => ( ( ord_less_real @ X6 @ B )
             => ? [Y2: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y2 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10166_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_10167_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X6: real] :
              ( ( ord_less_real @ A @ X6 )
             => ( ( ord_less_real @ X6 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X )
           => ( ( ord_less_eq_real @ X @ B )
             => ( ( F @ X )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10168_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 )
         => ( ! [X6: real] :
                ( ( ord_less_real @ A @ X6 )
               => ( ( ord_less_real @ X6 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X6 @ top_top_set_real ) ) ) )
           => ? [Z5: real] :
                ( ( ord_less_real @ A @ Z5 )
                & ( ord_less_real @ Z5 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10169_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X3: real,Y3: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X3 @ Y3 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10170_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X3: complex,Y3: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X3 @ Y3 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10171_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] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ N6 @ M3 )
           => ! [N2: nat] :
                ( ( ord_less_eq_nat @ N6 @ N2 )
               => ( P @ ( product_Pair_nat_nat @ N2 @ M3 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10172_mono__times__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).

% mono_times_nat
thf(fact_10173_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10174_incseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_10175_filtermap__at__right__shift,axiom,
    ! [D: real,A: real] :
      ( ( filtermap_real_real
        @ ^ [X3: real] : ( minus_minus_real @ X3 @ D )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
      = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).

% filtermap_at_right_shift
thf(fact_10176_incseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
       => ~ ! [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_10177_at__right__to__0,axiom,
    ! [A: real] :
      ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
      = ( filtermap_real_real
        @ ^ [X3: real] : ( plus_plus_real @ X3 @ A )
        @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_right_to_0
thf(fact_10178_at__right__to__top,axiom,
    ( ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) )
    = ( filtermap_real_real @ inverse_inverse_real @ at_top_real ) ) ).

% at_right_to_top
thf(fact_10179_at__top__to__right,axiom,
    ( at_top_real
    = ( filtermap_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% at_top_to_right
thf(fact_10180_filtermap__ln__at__right,axiom,
    ( ( filtermap_real_real @ ln_ln_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
    = at_bot_real ) ).

% filtermap_ln_at_right
thf(fact_10181_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10182_remdups__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% remdups_upt
thf(fact_10183_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_10184_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10185_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sorted_list_of_set_range
thf(fact_10186_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_10187_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_10188_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_10189_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_10190_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_10191_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10192_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10193_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10194_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I5: nat,J3: nat] : ( set_nat2 @ ( upt @ I5 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10195_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10196_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10197_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_10198_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_10199_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I5: nat] : ( plus_plus_nat @ I5 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_10200_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_10201_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_10202_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_10203_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10204_upt__rec,axiom,
    ( upt
    = ( ^ [I5: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I5 @ J3 ) @ ( cons_nat @ I5 @ ( upt @ ( suc @ I5 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10205_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_10206_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_10207_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X3: nat] : X3
          @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).

% sum_list_upt

% Helper facts (40)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X2: real] : ( P @ X2 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [Y2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ na ) ) @ ( vEBT_VEBT_low @ xa @ na ) ) ) @ maxs ) @ Y2 ) ).

%------------------------------------------------------------------------------