%------------------------------------------------------------------------------
% File     : ITP287^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00461_025495
% 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    : 0094_VEBT_SuccPredImperative_00461_025495 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11900 (5701 unt;1664 typ;   0 def)
%            Number of atoms       : 30093 (13392 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 135164 (3141   ~; 525   |;2038   &;118168   @)
%                                         (   0 <=>;11292  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   94 (   6 avg)
%            Number of types       :  208 ( 207 usr)
%            Number of type conns  : 6559 (6559   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1460 (1457 usr;  76 con; 0-8 aty)
%            Number of variables   : 28171 (3265   ^;23939   !; 967   ?;28171   :)
% 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 21:01:31.657
%------------------------------------------------------------------------------
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    rat: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

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

% Explicit typings (1457)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
    archim7802044766580827645g_real: real > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
    archim3151403230148437115or_rat: rat > int ).

thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
    archim6058952711729229775r_real: real > int ).

thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
    archim7778729529865785530nd_rat: rat > int ).

thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
    archim8280529875227126926d_real: real > int ).

thf(sy_c_Array__Time_Olen_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_len_VEBT_VEBTi: array_VEBT_VEBTi > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Onth_001_Eo,type,
    array_nth_o: array_o > nat > heap_Time_Heap_o ).

thf(sy_c_Array__Time_Onth_001t__Nat__Onat,type,
    array_nth_nat: array_nat > nat > heap_Time_Heap_nat ).

thf(sy_c_Array__Time_Onth_001t__Option__Ooption_It__Nat__Onat_J,type,
    array_nth_option_nat: array_option_nat > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Array__Time_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_nth_VEBT_VEBTi: array_VEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Array__Time_Oof__list_001_Eo,type,
    array_of_list_o: list_o > heap_T5660665574680485309rray_o ).

thf(sy_c_Array__Time_Oof__list_001t__Int__Oint,type,
    array_of_list_int: list_int > heap_T1346037964561226099ay_int ).

thf(sy_c_Array__Time_Oof__list_001t__Nat__Onat,type,
    array_of_list_nat: list_nat > heap_T3836121109492952855ay_nat ).

thf(sy_c_Array__Time_Oof__list_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_615059503499738533_VEBTi: list_VEBT_VEBTi > heap_T8822477325091257596_VEBTi ).

thf(sy_c_Array__Time_Oupd_001t__VEBT____BuildupMemImp__OVEBTi,type,
    array_upd_VEBT_VEBTi: nat > vEBT_VEBTi > array_VEBT_VEBTi > heap_T8822477325091257596_VEBTi ).

thf(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oentails,type,
    entails: assn > assn > $o ).

thf(sy_c_Assertions_Oex__assn_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    ex_ass463751140784270563_VEBTi: ( list_VEBT_VEBTi > assn ) > assn ).

thf(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

thf(sy_c_Assertions_Osnga__assn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    snga_assn_VEBT_VEBTi: array_VEBT_VEBTi > list_VEBT_VEBTi > assn ).

thf(sy_c_Automation_OFI,type,
    fi: list_P8527749157015355191n_assn > assn > assn > assn > assn > assn > $o ).

thf(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: list_P8527749157015355191n_assn > assn > assn > assn > $o ).

thf(sy_c_Binomial_Obinomial,type,
    binomial: nat > nat > nat ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Int__Oint,type,
    bit_bi6516823479961619367ts_int: ( nat > $o ) > int ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Uint32__Ouint32,type,
    bit_bi705532357378895591uint32: ( nat > $o ) > uint32 ).

thf(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_bi5746210779246519537l_num1: ( nat > $o ) > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: ( nat > $o ) > $o ).

thf(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: num > num > num ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
    bit_ri7632146776885996613nteger: code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Uint32__Ouint32,type,
    bit_ri6224792872505173163uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_ri1375673916561920181l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Uint32__Ouint32,type,
    bit_se6294004230839889034uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Code____Numeral__Ointeger,type,
    bit_se3928097537394005634nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Uint32__Ouint32,type,
    bit_se3964402333458159761uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se5176125413884933531l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Uint32__Ouint32,type,
    bit_se7025624438249859091uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4491814353640558621l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Uint32__Ouint32,type,
    bit_se2966626333419230250uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Code____Numeral__Ointeger,type,
    bit_se7788150548672797655nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Uint32__Ouint32,type,
    bit_se5742574853984576102uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se837345729053750000l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Uint32__Ouint32,type,
    bit_se6647067497041451410uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se4894374433684937756l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Uint32__Ouint32,type,
    bit_se4315839071623982667uint32: nat > uint32 > uint32 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se5331074070815623765l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Code____Numeral__Ointeger,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Uint32__Ouint32,type,
    bit_se5367290876889521763uint32: uint32 > nat > $o ).

thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    bit_se6859397288646540909l_num1: word_N3645301735248828278l_num1 > nat > $o ).

thf(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: nat > num > option_num ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl_001t__Nat__Onat,type,
    bit_Sh3965577149348748681tl_nat: nat > nat > nat ).

thf(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr_001t__Nat__Onat,type,
    bit_Sh2154871086232339855tr_nat: nat > nat > nat ).

thf(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: code_integer > $o > code_integer ).

thf(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

thf(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

thf(sy_c_Bits__Integer_Ointeger__set__bit,type,
    bits_integer_set_bit: code_integer > code_integer > $o > code_integer ).

thf(sy_c_Bits__Integer_Ointeger__shiftl,type,
    bits_integer_shiftl: code_integer > code_integer > code_integer ).

thf(sy_c_Bits__Integer_Ointeger__shiftr,type,
    bits_integer_shiftr: code_integer > code_integer > code_integer ).

thf(sy_c_Code__Int__Integer__Conversion_Oint__of__integer__symbolic,type,
    code_I935103866777955880mbolic: code_integer > int ).

thf(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Code__Numeral_Odup,type,
    code_dup: code_integer > code_integer ).

thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

thf(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

thf(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

thf(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: num > int ).

thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Int__Oint,type,
    code_T3171777517611823113le_int: ( code_integer > product_prod_int_int ) > int > code_integer > int ).

thf(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample_001t__Nat__Onat,type,
    code_T3174267988120873389le_nat: ( code_integer > product_prod_nat_nat ) > nat > code_integer > nat ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple2654586775044187945Heap_o: ( set_Pr2007700399681132348Heap_o > produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple1540308706681863803on_nat: ( set_Pr1591120925906170302on_nat > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple1745167176254620304_VEBTi: ( set_Pr2840599766253930323_VEBTi > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple6491863954676465222Heap_o: ( set_Pr5371233824415811545Heap_o > produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple6931689918642796574on_nat: ( set_Pr6126824603708961249on_nat > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple380401974140132787_VEBTi: ( set_Pr3980204975930894582_VEBTi > produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > ( ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > $o ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple3202505432650402847Heap_o: ( set_Pr2007700399681132348Heap_o > produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > produc3960855945107176009Ti_nat > heap_Time_Heap_o ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple8068445680736955397on_nat: ( set_Pr1591120925906170302on_nat > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    comple7072962176332223770_VEBTi: ( set_Pr2840599766253930323_VEBTi > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    comple2405882057716616508Heap_o: ( set_Pr5371233824415811545Heap_o > produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > produc3881548065746020326Ti_nat > heap_Time_Heap_o ).

thf(sy_c_Complete__Partial__Order_Occpo_Ofixp_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    comple6805837186910174120on_nat: ( set_Pr6126824603708961249on_nat > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple6074371103668693207Heap_o: ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o ) > ( heap_Time_Heap_o > heap_Time_Heap_o > $o ) > ( ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    comple6977564771798581627on_nat: ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple5606513277678308283_VEBTi: ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > ( heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ) > ( ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple4217288648910406772Heap_o: ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > ( heap_Time_Heap_o > heap_Time_Heap_o > $o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    comple3826860765959394442ap_nat: ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > ( heap_Time_Heap_nat > heap_Time_Heap_nat > $o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_nat ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple5335682857743707887_VEBTi: ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ) > ( heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ) > ( ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    comple6677746081827660726Heap_o: ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( heap_Time_Heap_o > heap_Time_Heap_o > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    comple1015018851985181128ap_nat: ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( heap_Time_Heap_nat > heap_Time_Heap_nat > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_nat ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    comple4655144769394346904on_nat: ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple2969382418784824877_VEBTi: ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o ) > ( heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi ) > $o ).

thf(sy_c_Complete__Partial__Order_Omonotone_001_062_It__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    comple2284608890766496472_VEBTi: ( ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > $o ) > ( heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ) > ( ( produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ) > $o ).

thf(sy_c_Complex_OArg,type,
    arg: complex > real ).

thf(sy_c_Complex_Ocis,type,
    cis: real > complex ).

thf(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > real > complex ).

thf(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
    semiri2265585572941072030t_real: nat > real ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
    invers8013647133539491842omplex: complex > complex ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
    inverse_inverse_rat: rat > rat ).

thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
    at_top_real: filter_real ).

thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
    finite_finite_o: set_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Code____Numeral__Ointeger,type,
    finite6017078050557962740nteger: set_Code_integer > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
    finite3207457112153483333omplex: set_complex > $o ).

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

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
    finite_finite_list_o: set_list_o > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    finite1283093830868386564nteger: set_li6976499617229504675nteger > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    finite1351478129840809056n_assn: set_li5131720305576846103n_assn > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Real__Oreal_J,type,
    finite306553202115118035t_real: set_list_real > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

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

thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
    finite_finite_num: set_num > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    finite5137929494490007386n_assn: set_Pr5949110396991348497n_assn > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Int__Oint_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    groups8925579862173457374l_num1: ( int > word_N3645301735248828278l_num1 ) > set_int > word_N3645301735248828278l_num1 ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Rat__Orat,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Int__Oint,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Nat__Onat,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Rat__Orat,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001_Eo_001t__Int__Oint,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(sy_c_HOL_OThe_001t__Int__Oint,type,
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thf(sy_c_HOL_OThe_001t__Real__Oreal,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_I_062_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J_M_062_It__Uint32__Ouint32_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Code____Numeral__Ointeger_J_J_J,type,
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thf(sy_c_HOL_Oundefined_001_062_It__Code____Numeral__Ointeger_M_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
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thf(sy_c_HOL_Oundefined_001t__Int__Oint,type,
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thf(sy_c_HOL_Oundefined_001t__Nat__Onat,type,
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thf(sy_c_HOL_Oundefined_001t__Num__Onum,type,
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thf(sy_c_HOL_Oundefined_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_HOL_Oundefined_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
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thf(sy_c_Heap_Oarray_Osize__array_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Heap__Time__Monad_OHeap_OHeap_001_Eo,type,
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thf(sy_c_Heap__Time__Monad_OHeap_OHeap_001t__Nat__Onat,type,
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thf(sy_c_Heap__Time__Monad_OHeap_OHeap_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T5286843759275942675on_nat: ( heap_e7401611519738050253t_unit > option2621746655072343315it_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_OHeap_OHeap_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T1489671670754571048_VEBTi: ( heap_e7401611519738050253t_unit > option5408194888911472936it_nat ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_OHeap__lub_001_Eo,type,
    heap_Time_Heap_lub_o: set_Heap_Time_Heap_o > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_OHeap__lub_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Heap__Time__Monad_OHeap__lub_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T3112222404744780921_VEBTi: set_He5684063546058238497_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_OHeap__ord_001_Eo,type,
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thf(sy_c_Heap__Time__Monad_OHeap__ord_001t__Nat__Onat,type,
    heap_T645984214031783516rd_nat: heap_Time_Heap_nat > heap_Time_Heap_nat > $o ).

thf(sy_c_Heap__Time__Monad_OHeap__ord_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T7875578835903804844on_nat: heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > $o ).

thf(sy_c_Heap__Time__Monad_OHeap__ord_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Heap__Time__Monad_Obind_001_Eo_001_Eo,type,
    heap_Time_bind_o_o: heap_Time_Heap_o > ( $o > heap_Time_Heap_o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Obind_001_Eo_001t__Nat__Onat,type,
    heap_Time_bind_o_nat: heap_Time_Heap_o > ( $o > heap_Time_Heap_nat ) > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001_Eo_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Heap__Time__Monad_Obind_001_Eo_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T5998771940306268294_VEBTi: heap_Time_Heap_o > ( $o > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T5099337393651448672_VEBTi: heap_T8822477325091257596_VEBTi > ( array_VEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T5877712393672139267_VEBTi: heap_T4980287057938770641_VEBTi > ( list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001_Eo,type,
    heap_Time_bind_nat_o: heap_Time_Heap_nat > ( nat > heap_Time_Heap_o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    heap_T7049098217575491753at_nat: heap_Time_Heap_nat > ( nat > heap_Time_Heap_nat ) > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T8222160169144143993on_nat: heap_Time_Heap_nat > ( nat > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T844888390831797134_VEBTi: heap_Time_Heap_nat > ( nat > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Option__Ooption_It__Nat__Onat_J_001_Eo,type,
    heap_T6471384023045698863_nat_o: heap_T2636463487746394924on_nat > ( option_nat > heap_Time_Heap_o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T3669509953089699273on_nat: heap_T2636463487746394924on_nat > ( option_nat > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Option__Ooption_It__Nat__Onat_J_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T5661892481228163294_VEBTi: heap_T2636463487746394924on_nat > ( option_nat > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001_Eo,type,
    heap_T7577940988442287570unit_o: heap_T5738788834812785303t_unit > ( product_unit > heap_Time_Heap_o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    heap_T3781436268274291734it_nat: heap_T5738788834812785303t_unit > ( product_unit > heap_Time_Heap_nat ) > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T5999496708990702694on_nat: heap_T5738788834812785303t_unit > ( product_unit > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__Product____Type__Ounit_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T7982501707604696571_VEBTi: heap_T5738788834812785303t_unit > ( product_unit > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Obind_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    heap_T3040810144269856602EBTi_o: heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > heap_Time_Heap_o ) > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Obind_001t__VEBT____BuildupMemImp__OVEBTi_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T2868974464944644318on_nat: heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > heap_T2636463487746394924on_nat ) > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Obind_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T1006145433769338483_VEBTi: heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Oeffect_001_Eo,type,
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thf(sy_c_Heap__Time__Monad_Oeffect_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    heap_T33481931004607297_VEBTi: heap_T4980287057938770641_VEBTi > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > list_VEBT_VEBTi > nat > $o ).

thf(sy_c_Heap__Time__Monad_Oeffect_001t__Nat__Onat,type,
    heap_Time_effect_nat: heap_Time_Heap_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > nat > nat > $o ).

thf(sy_c_Heap__Time__Monad_Oeffect_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T306965388786959644on_nat: heap_T2636463487746394924on_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o ).

thf(sy_c_Heap__Time__Monad_Oeffect_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T2071195472996403633_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o ).

thf(sy_c_Heap__Time__Monad_Oreturn_001_Eo,type,
    heap_Time_return_o: $o > heap_Time_Heap_o ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Nat__Onat,type,
    heap_Time_return_nat: nat > heap_Time_Heap_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Option__Ooption_It__Nat__Onat_J,type,
    heap_T3487192422709364219on_nat: option_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__Product____Type__Ounit,type,
    heap_T7507251653302230130t_unit: product_unit > heap_T5738788834812785303t_unit ).

thf(sy_c_Heap__Time__Monad_Oreturn_001t__VEBT____BuildupMemImp__OVEBTi,type,
    heap_T3630416162098727440_VEBTi: vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_T5738788834812785303t_unit ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001_Eo,type,
    hoare_hoare_triple_o: assn > heap_Time_Heap_o > ( $o > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Nat__Onat,type,
    hoare_3067605981109127869le_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Product____Type__Ounit,type,
    hoare_8945653483474564448t_unit: assn > heap_T5738788834812785303t_unit > ( product_unit > assn ) > $o ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__VEBT____BuildupMemImp__OVEBTi,type,
    hoare_1429296392585015714_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > $o ).

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

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

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

thf(sy_c_If_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_If_001t__Heap____Time____Monad__OHeap_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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thf(sy_c_If_001t__Int__Oint,type,
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thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Num__Onum,type,
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thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J,type,
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thf(sy_c_If_001t__Rat__Orat,type,
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thf(sy_c_If_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__Uint32__Ouint32,type,
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thf(sy_c_If_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_If_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
    ring_17405671764205052669omplex: int > complex ).

thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Uint32__Ouint32,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ring_17408606157368542149l_num1: int > word_N3645301735248828278l_num1 ).

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

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Least__significant__bit_Olsb__class_Olsb_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001_Eo,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oenumerate_001_Eo,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oenumerate_001t__Num__Onum,type,
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thf(sy_c_List_Oenumerate_001t__Real__Oreal,type,
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thf(sy_c_List_Oenumerate_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001_Eo_001t__Nat__Onat,type,
    foldr_o_nat: ( $o > nat > nat ) > list_o > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_List_Ofoldr_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Nat__Onat_001t__Nat__Onat,type,
    foldr_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Nat__Onat,type,
    foldr_real_nat: ( real > nat > nat ) > list_real > nat > nat ).

thf(sy_c_List_Ofoldr_001t__Real__Oreal_001t__Real__Oreal,type,
    foldr_real_real: ( real > real > real ) > list_real > real > real ).

thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
    linord1735203802627413978nt_int: ( int > int ) > list_int > list_int ).

thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001_Eo,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
    cons_int: int > list_int > list_int ).

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

thf(sy_c_List_Olist_ONil_001_Eo,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
    map_o_o: ( $o > $o ) > list_o > list_o ).

thf(sy_c_List_Olist_Omap_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001_Eo,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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    map_Pr454908937103039467um_num: ( product_prod_num_num > num ) > list_P3744719386663036955um_num > list_num ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J_001t__VEBT____Definitions__OVEBT,type,
    map_Pr6147841162850987569T_VEBT: ( produc3757001726724277373T_VEBT > vEBT_VEBT ) > list_P877281246627933069T_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_001t__VEBT____Definitions__OVEBT,type,
    map_Pr4868735216952053677T_VEBT: ( produc334124729049499915VEBT_o > vEBT_VEBT ) > list_P3126845725202233233VEBT_o > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_001t__VEBT____Definitions__OVEBT,type,
    map_Pr3257657825534036127T_VEBT: ( produc4894624898956917775BT_int > vEBT_VEBT ) > list_P4547456442757143711BT_int > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_001t__VEBT____Definitions__OVEBT,type,
    map_Pr1380729192516676091T_VEBT: ( produc9072475918466114483BT_nat > vEBT_VEBT ) > list_P7037539587688870467BT_nat > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J_001t__VEBT____Definitions__OVEBT,type,
    map_Pr6195879527588727455T_VEBT: ( produc5170161368751668367T_real > vEBT_VEBT ) > list_P2623026923184700063T_real > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001_Eo,type,
    map_real_o: ( real > $o ) > list_real > list_o ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
    map_real_nat: ( real > nat ) > list_real > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    map_re7205069664741861231T_VEBT: ( real > produc3757001726724277373T_VEBT ) > list_real > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    map_re8618229306769252225T_real: ( real > produc5170161368751668367T_real ) > list_real > list_P2623026923184700063T_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
    map_real_real: ( real > real ) > list_real > list_real ).

thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    map_real_VEBT_VEBT: ( real > vEBT_VEBT ) > list_real > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Int__Oint,type,
    map_VEBT_VEBTi_int: ( vEBT_VEBTi > int ) > list_VEBT_VEBTi > list_int ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat,type,
    map_VEBT_VEBTi_nat: ( vEBT_VEBTi > nat ) > list_VEBT_VEBTi > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE483055756984248624_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi ) > list_VEBT_VEBTi > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    map_VE7998069337340375161T_VEBT: ( vEBT_VEBTi > vEBT_VEBT ) > list_VEBT_VEBTi > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    map_VEBT_VEBT_int: ( vEBT_VEBT > int ) > list_VEBT_VEBT > list_int ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    map_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > list_nat ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    map_VEBT_VEBT_real: ( vEBT_VEBT > real ) > list_VEBT_VEBT > list_real ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    map_VE7029150624388687525_VEBTi: ( vEBT_VEBT > vEBT_VEBTi ) > list_VEBT_VEBT > list_VEBT_VEBTi ).

thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Olist_Oset_001_Eo,type,
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thf(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
    set_Code_integer2: list_Code_integer > set_Code_integer ).

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

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    set_Pr1139785259514867910n_assn: list_P8527749157015355191n_assn > set_Pr5949110396991348497n_assn ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    set_Pr1291962091234853352_nat_o: list_P7333126701944960589_nat_o > set_Pr3149072824959771635_nat_o ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    set_Pr1470767568048878706at_int: list_P3521021558325789923at_int > set_Pr7995236796853374141at_int ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    set_Pr7149346036329476978t_real: list_P3644420460460130531t_real > set_Pr320017278500174781t_real ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    set_Pr4207466110102731387_VEBTi: list_P2320588648998582380_VEBTi > set_Pr1938536134445252166_VEBTi ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    set_Pr5984661752051438084T_VEBT: list_P5647936690300460905T_VEBT > set_Pr6167073792073659919T_VEBT ).

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

thf(sy_c_List_Olist_Oset_001t__VEBT____BuildupMemImp__OVEBTi,type,
    set_VEBT_VEBTi2: list_VEBT_VEBTi > set_VEBT_VEBTi ).

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__Int__Oint,type,
    list_update_int: list_int > nat > int > list_int ).

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    list_u4534839942911652127n_assn: list_P8527749157015355191n_assn > nat > produc6575502325842934193n_assn > list_P8527749157015355191n_assn ).

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

thf(sy_c_List_Olist__update_001t__VEBT____BuildupMemImp__OVEBTi,type,
    list_u6098035379799741383_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi > list_VEBT_VEBTi ).

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__Int__Oint,type,
    nth_int: list_int > nat > int ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    nth_Pr1769885009046257848n_assn: list_P8527749157015355191n_assn > nat > produc6575502325842934193n_assn ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr2419613052044807976T_VEBT: list_P6730324909620535719T_VEBT > nat > produc4813437837504472865T_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__Int__Oint_J,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    nth_Pr5469784954002723455T_VEBT: list_P4737670876410327766T_VEBT > nat > produc8398139464844984134T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    nth_Pr7767817059697521252t_real: list_P3644420460460130531t_real > nat > produc7716430852924023517t_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr3244165891152107629_VEBTi: list_P2320588648998582380_VEBTi > nat > produc214224863196444774_VEBTi ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_M_Eo_J,type,
    nth_Pr3306050735993963089EBTi_o: list_P8833571063612306856EBTi_o > nat > produc5014006835512566296EBTi_o ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Real__Oreal_J,type,
    nth_Pr3433448822664029129i_real: list_P8536626330812492744i_real > nat > produc6680258955013199682i_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr6329974346453275474_VEBTi: list_P785718909624839377_VEBTi > nat > produc3777764054643897931_VEBTi ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr8725177398587324397T_VEBT: list_P5988454224134618948T_VEBT > nat > produc2810682830582626868T_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__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__Real__Oreal_J,type,
    nth_Pr6842391030413306568T_real: list_P2623026923184700063T_real > nat > produc5170161368751668367T_real ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    nth_Pr316670251186196177_VEBTi: list_P735349106241217576_VEBTi > nat > produc3625547720036274456_VEBTi ).

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

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

thf(sy_c_List_Onth_001t__VEBT____BuildupMemImp__OVEBTi,type,
    nth_VEBT_VEBTi: list_VEBT_VEBTi > nat > vEBT_VEBTi ).

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__Real__Oreal,type,
    product_o_real: list_o > list_real > list_P5232166724548748803o_real ).

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__Real__Oreal,type,
    product_nat_real: list_nat > list_real > list_P3644420460460130531t_real ).

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

thf(sy_c_List_Oproduct_001t__Real__Oreal_001_Eo,type,
    product_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Int__Oint,type,
    product_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Nat__Onat,type,
    product_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Oproduct_001t__Real__Oreal_001t__Real__Oreal,type,
    product_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001_Eo,type,
    product_VEBT_VEBTi_o: list_VEBT_VEBTi > list_o > list_P8833571063612306856EBTi_o ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__Real__Oreal,type,
    produc5476717833281694120i_real: list_VEBT_VEBTi > list_real > list_P8536626330812492744i_real ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc194614972289024177_VEBTi: list_VEBT_VEBTi > list_VEBT_VEBTi > list_P785718909624839377_VEBTi ).

thf(sy_c_List_Oproduct_001t__VEBT____BuildupMemImp__OVEBTi_001t__VEBT____Definitions__OVEBT,type,
    produc1285381384045549624T_VEBT: list_VEBT_VEBTi > list_VEBT_VEBT > list_P5988454224134618948T_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__Nat__Onat,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc4908677263432625371T_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    produc316462671093861988_VEBTi: list_VEBT_VEBT > list_VEBT_VEBTi > list_P735349106241217576_VEBTi ).

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

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

thf(sy_c_List_Oreplicate_001t__Assertions__Oassn,type,
    replicate_assn: nat > assn > list_assn ).

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__Num__Onum,type,
    replicate_num: nat > num > list_num ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    replic3825545231534752113n_assn: nat > produc6575502325842934193n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    replic862811827245231841T_VEBT: nat > produc4813437837504472865T_VEBT > list_P6730324909620535719T_VEBT ).

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

thf(sy_c_List_Oreplicate_001t__VEBT____BuildupMemImp__OVEBTi,type,
    replicate_VEBT_VEBTi: nat > vEBT_VEBTi > list_VEBT_VEBTi ).

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_Misc_Orel__of_001t__Int__Oint_001t__Int__Oint,type,
    rel_of_int_int: ( int > option_int ) > ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Nat__Onat,type,
    rel_of_nat_nat: ( nat > option_nat ) > ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Num__Onum,type,
    rel_of_nat_num: ( nat > option_num ) > ( product_prod_nat_num > $o ) > set_Pr6200539531224447659at_num ).

thf(sy_c_Misc_Orel__of_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    rel_of7001861737123029207T_VEBT: ( nat > option1280308654545898343T_VEBT ) > ( produc8398139464844984134T_VEBT > $o ) > set_Pr563407847431865468T_VEBT ).

thf(sy_c_Misc_Orel__of_001t__Num__Onum_001t__Num__Onum,type,
    rel_of_num_num: ( num > option_num ) > ( product_prod_num_num > $o ) > set_Pr8218934625190621173um_num ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Code____Numeral__Ointeger,type,
    most_s5661112943643946085nteger: code_integer > $o ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Int__Oint,type,
    most_s5051101344085556sb_int: int > $o ).

thf(sy_c_Most__significant__bit_Omsb__class_Omsb_001t__Uint32__Ouint32,type,
    most_s9063628576841037300uint32: uint32 > $o ).

thf(sy_c_Nat_OSuc,type,
    suc: 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_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
    semiri4939895301339042750nteger: nat > code_integer ).

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

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

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

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: nat > rat ).

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

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Uint32__Ouint32,type,
    semiri2565882477558803405uint32: nat > uint32 ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    semiri8819519690708144855l_num1: nat > word_N3645301735248828278l_num1 ).

thf(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    size_s2700093152935483318Heap_o: heap_Time_Heap_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    size_s6287829766004316056on_nat: heap_T2636463487746394924on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s8425857057747876397_VEBTi: heap_T8145700208782473153_VEBTi > nat ).

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__Assertions__Oassn_J,type,
    size_size_list_assn: list_assn > nat ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
    size_s2624279037499656343o_real: list_P5232166724548748803o_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    size_s6829681357464350627n_assn: list_P8527749157015355191n_assn > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    size_s4764337671732037139T_VEBT: list_P6730324909620535719T_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__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    size_s7910714270633306959t_real: list_P3644420460460130531t_real > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
    size_s987546567493390085real_o: list_P3595434254542482545real_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
    size_s8610625264895183403al_int: list_P4344331454722006975al_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    size_s1877336372972134351al_nat: list_P6834414599653733731al_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    size_s3932428310213730859l_real: list_P8689742595348180415l_real > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    size_s7982070591426661849_VEBTi: list_VEBT_VEBTi > 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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s364044314319911927it_nat: option7339022715339332451it_nat > 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__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
    size_s3991424295186984831it_nat: option2621746655072343315it_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__Uint32__Ouint32,type,
    size_size_uint32: uint32 > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____BuildupMemImp__OVEBTi,type,
    size_size_VEBT_VEBTi: vEBT_VEBTi > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    size_s8261804613246490634l_num1: word_N3645301735248828278l_num1 > 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__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_001t__Uint32__Ouint32,type,
    neg_nu5314729912787363643uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    neg_nu7865238048354675525l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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__dec_001t__Uint32__Ouint32,type,
    neg_nu965353292909893953uint32: uint32 > uint32 ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    neg_nu93272222329896523l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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_Onumeral__class_Onumeral_001t__Uint32__Ouint32,type,
    numera9087168376688890119uint32: num > uint32 ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    numera7442385471795722001l_num1: num > word_N3645301735248828278l_num1 ).

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

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

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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7668321371905463026it_nat: option7339022715339332451it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    none_P2377608414092835994nt_int: option4624381673175914239nt_int ).

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_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    none_P6264349658649815852at_num: option642762832853965969at_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P281974696781278558it_nat: option7211493094183709123it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    none_P330983522480640549T_VEBT: option254855292876462358T_VEBT ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    none_P4394680061957285238um_num: option2661157926820139483um_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P1551326421579882414it_nat: option2621746655072343315it_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    none_P7832717587476222275it_nat: option5408194888911472936it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P468703482102919278it_nat: produc8047831477865546771it_nat > option7339022715339332451it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    some_P4184893108420464158nt_int: product_prod_int_int > option4624381673175914239nt_int ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    some_P2407225848856114310T_VEBT: produc4813437837504472865T_VEBT > option1280308654545898343T_VEBT ).

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_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    some_P8071634352977444016at_num: product_prod_nat_num > option642762832853965969at_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    some_P5996745733903548321T_VEBT: produc8398139464844984134T_VEBT > option254855292876462358T_VEBT ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    some_P6201964756284913402um_num: product_prod_num_num > option2661157926820139483um_num ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    some_P2407035485129114418it_nat: produc120671012495760973it_nat > option2621746655072343315it_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Rat__Orat,type,
    some_rat: rat > option_rat ).

thf(sy_c_Option_Ooption_OSome_001t__Real__Oreal,type,
    some_real: real > option_real ).

thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__Nat__Onat_J,type,
    some_set_nat: set_nat > option_set_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    case_o8336680350232271869uint32: ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) > ( ( ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) > ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ) > option8496191915386069960uint32 > ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    case_o6516889040143735037uint32: ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) > ( ( ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) > ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ) > option373713263958016584uint32 > ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    case_o6228893485755354685uint32: ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) > ( ( ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) > ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ) > option7887515136451277736uint32 > ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    case_o4437601675458612413eger_o: ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) > ( ( ( uint32 > nat > $o ) > uint32 > code_integer > $o ) > ( uint32 > nat > $o ) > uint32 > code_integer > $o ) > option4062567599839601128eger_o > ( uint32 > nat > $o ) > uint32 > code_integer > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_062_I_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J_001_062_I_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    case_o6709414378691970003uint32: ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) > ( ( ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) > ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ) > option8540941645471956339uint32 > ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Int__Oint,type,
    case_option_o_int: $o > ( int > $o ) > option_int > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Nat__Onat,type,
    case_option_o_nat: $o > ( nat > $o ) > option_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Num__Onum,type,
    case_option_o_num: $o > ( num > $o ) > option_num > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o1358941076187788256it_nat: $o > ( produc8047831477865546771it_nat > $o ) > option7339022715339332451it_nat > $o ).

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_001_Eo_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    case_o535201446637900608it_nat: $o > ( produc120671012495760973it_nat > $o ) > option2621746655072343315it_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Rat__Orat,type,
    case_option_o_rat: $o > ( rat > $o ) > option_rat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Real__Oreal,type,
    case_option_o_real: $o > ( real > $o ) > option_real > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    case_o4401850862724306899et_nat: $o > ( set_nat > $o ) > option_set_nat > $o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Nat__Onat,type,
    case_o6892868863119666303_o_nat: heap_Time_Heap_o > ( nat > heap_Time_Heap_o ) > option_nat > heap_Time_Heap_o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Num__Onum,type,
    case_o3450200649275444937_o_num: heap_Time_Heap_o > ( num > heap_Time_Heap_o ) > option_num > heap_Time_Heap_o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1442776274061689234at_nat: heap_Time_Heap_o > ( product_prod_nat_nat > heap_Time_Heap_o ) > option4927543243414619207at_nat > heap_Time_Heap_o ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Nat__Onat,type,
    case_o6609685678014844897at_nat: heap_Time_Heap_nat > ( nat > heap_Time_Heap_nat ) > option_nat > heap_Time_Heap_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Num__Onum,type,
    case_o3167017464170623531at_num: heap_Time_Heap_nat > ( num > heap_Time_Heap_nat ) > option_num > heap_Time_Heap_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o3959993630158478256at_nat: heap_Time_Heap_nat > ( product_prod_nat_nat > heap_Time_Heap_nat ) > option4927543243414619207at_nat > heap_Time_Heap_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Nat__Onat,type,
    case_o2256915875499652529at_nat: heap_T2636463487746394924on_nat > ( nat > heap_T2636463487746394924on_nat ) > option_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Num__Onum,type,
    case_o8037619698510206971at_num: heap_T2636463487746394924on_nat > ( num > heap_T2636463487746394924on_nat ) > option_num > heap_T2636463487746394924on_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o8344607093967974880at_nat: heap_T2636463487746394924on_nat > ( product_prod_nat_nat > heap_T2636463487746394924on_nat ) > option4927543243414619207at_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat,type,
    case_o3780387683879180358Ti_nat: heap_T8145700208782473153_VEBTi > ( nat > heap_T8145700208782473153_VEBTi ) > option_nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Num__Onum,type,
    case_o337719470034958992Ti_num: heap_T8145700208782473153_VEBTi > ( num > heap_T8145700208782473153_VEBTi ) > option_num > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o1356590567247012107at_nat: heap_T8145700208782473153_VEBTi > ( product_prod_nat_nat > heap_T8145700208782473153_VEBTi ) > option4927543243414619207at_nat > heap_T8145700208782473153_VEBTi ).

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__Nat__Onat_001t__Nat__Onat,type,
    case_option_nat_nat: nat > ( nat > nat ) > option_nat > nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Num__Onum,type,
    case_option_nat_num: nat > ( num > nat ) > option_num > nat ).

thf(sy_c_Option_Ooption_Ocase__option_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o2098746482150326116at_nat: nat > ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

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_Ocase__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o7430979018509204427at_nat: product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > product_prod_nat_nat ).

thf(sy_c_Option_Ooption_Othe_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Code____Numeral__Ointeger_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    the_na2292640131888687716uint32: option8496191915386069960uint32 > ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32 ).

thf(sy_c_Option_Ooption_Othe_001_062_I_062_It__Nat__Onat_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_Mt__Uint32__Ouint32_J_J_J,type,
    the_na3915024202274359524uint32: option373713263958016584uint32 > ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32 ).

thf(sy_c_Option_Ooption_Othe_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_062_I_Eo_Mt__Uint32__Ouint32_J_J_J_J,type,
    the_ui8720505876773817540uint32: option7887515136451277736uint32 > ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32 ).

thf(sy_c_Option_Ooption_Othe_001_062_I_062_It__Uint32__Ouint32_M_062_It__Nat__Onat_M_Eo_J_J_M_062_It__Uint32__Ouint32_M_062_It__Code____Numeral__Ointeger_M_Eo_J_J_J,type,
    the_ui5136145761085816068eger_o: option4062567599839601128eger_o > ( uint32 > nat > $o ) > uint32 > code_integer > $o ).

thf(sy_c_Option_Ooption_Othe_001_062_I_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_M_062_It__Uint32__Ouint32_M_062_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_J_J,type,
    the_ui685118366354182287uint32: option8540941645471956339uint32 > ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32 ).

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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr5838048819577852031it_nat: option7339022715339332451it_nat > produc8047831477865546771it_nat ).

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_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    the_Pr3501439614016493281it_nat: option2621746655072343315it_nat > produc120671012495760973it_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bot_bot_int_int_o: int > int > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_nat_nat_o: nat > nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    bot_bot_nat_num_o: nat > num > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_M_Eo_J_J,type,
    bot_bo7529698899530922655VEBT_o: nat > produc4813437837504472865T_VEBT > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J,type,
    bot_bot_num_num_o: num > num > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

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__Option__Ooption_It__Nat__Onat_J,type,
    bot_bot_option_nat: option_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Option__Ooption_It__Num__Onum_J,type,
    bot_bot_option_num: option_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    bot_bo3990330152332043303nteger: set_Code_integer ).

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__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
    bot_bo3236126332025433324Heap_o: set_Heap_Time_Heap_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
    bot_bo8932748503833948152on_nat: set_He5367250461312314764on_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
    bot_bo3125955617464001165_VEBTi: set_He5684063546058238497_VEBTi ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    bot_bo1176836662018730877n_assn: set_Pr5949110396991348497n_assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    bot_bo7038385379056416535at_num: set_Pr6200539531224447659at_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    bot_bo9115540109607619856T_VEBT: set_Pr563407847431865468T_VEBT ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    bot_bo9056780473022590049um_num: set_Pr8218934625190621173um_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__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_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    ord_less_VEBT_VEBT_o: ( vEBT_VEBT > $o ) > ( vEBT_VEBT > $o ) > $o ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_less_option_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_less_option_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_less_option_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_less_option_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Option__Ooption_It__Real__Oreal_J,type,
    ord_less_option_real: option_real > option_real > $o ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_less_set_uint32: set_uint32 > set_uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    ord_le3480810397992357184T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_J,type,
    ord_le6726900395242856064l_num1: set_wo3913738467083021356l_num1 > set_wo3913738467083021356l_num1 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__String__Ochar,type,
    ord_less_char: char > char > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Uint32__Ouint32,type,
    ord_less_uint32: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le750835935415966154l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $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__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    ord_le3404735783095501756_num_o: ( nat > num > $o ) > ( nat > num > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_M_Eo_J_J,type,
    ord_le870442331779451499VEBT_o: ( nat > produc4813437837504472865T_VEBT > $o ) > ( nat > produc4813437837504472865T_VEBT > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J,type,
    ord_le6124364862034508274_num_o: ( num > num > $o ) > ( num > num > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Assertions__Oassn,type,
    ord_less_eq_assn: assn > assn > $o ).

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Int__Oint_J,type,
    ord_le1736525451366464988on_int: option_int > option_int > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Nat__Onat_J,type,
    ord_le5914376470875661696on_nat: option_nat > option_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Num__Onum_J,type,
    ord_le6622620407824499402on_num: option_num > option_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Rat__Orat_J,type,
    ord_le2406147912482264968on_rat: option_rat > option_rat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Option__Ooption_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le2843612097646854710et_nat: option_set_nat > option_set_nat > $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__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    ord_le171416862856029873n_assn: set_Pr5949110396991348497n_assn > set_Pr5949110396991348497n_assn > $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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    ord_le8085105155179020875at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    ord_le6438908469242860764T_VEBT: set_Pr563407847431865468T_VEBT > set_Pr563407847431865468T_VEBT > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    ord_le880128212290418581um_num: set_Pr8218934625190621173um_num > set_Pr8218934625190621173um_num > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Uint32__Ouint32_J,type,
    ord_le2219237028632753026uint32: set_uint32 > set_uint32 > $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_Oless__eq_001t__Set__Oset_It__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_J,type,
    ord_le5203802739334966412l_num1: set_wo3913738467083021356l_num1 > set_wo3913738467083021356l_num1 > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__String__Ochar,type,
    ord_less_eq_char: char > char > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Uint32__Ouint32,type,
    ord_less_eq_uint32: uint32 > uint32 > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_le3335648743751981014l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Uint32__Ouint32,type,
    ord_max_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    ord_ma8239519435860878689l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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

thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

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

thf(sy_c_Orderings_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__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_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6726927458685305659Ti_nat: ( set_Heap_Time_Heap_o > heap_Time_Heap_o ) > set_Pr2007700399681132348Heap_o > produc3960855945107176009Ti_nat > heap_Time_Heap_o ).

thf(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia5551857090987368152Ti_nat: ( set_Heap_Time_Heap_o > heap_Time_Heap_o ) > set_Pr5371233824415811545Heap_o > produc3881548065746020326Ti_nat > heap_Time_Heap_o ).

thf(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia6039416512482706817Ti_nat: ( set_He5367250461312314764on_nat > heap_T2636463487746394924on_nat ) > set_Pr1591120925906170302on_nat > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia2080987842261039902Ti_nat: ( set_He5367250461312314764on_nat > heap_T2636463487746394924on_nat ) > set_Pr6126824603708961249on_nat > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia7782936097874681665Ti_nat: ( set_He5684063546058238497_VEBTi > heap_T8145700208782473153_VEBTi ) > set_Pr2840599766253930323_VEBTi > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Partial__Function_Ofun__lub_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia6972460264168101086Ti_nat: ( set_He5684063546058238497_VEBTi > heap_T8145700208782473153_VEBTi ) > set_Pr3980204975930894582_VEBTi > produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_Mt__Nat__Onat_J,type,
    partia3290229181235258227Ti_nat: ( heap_Time_Heap_o > heap_Time_Heap_o > $o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o ).

thf(sy_c_Partial__Function_Ofun__ord_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Heap____Time____Monad__OHeap_I_Eo_J_001t__Product____Type__Oprod_It__VEBT____BuildupMemImp__OVEBTi_Mt__Nat__Onat_J,type,
    partia2925774515620677392Ti_nat: ( heap_Time_Heap_o > heap_Time_Heap_o > $o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o ).

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thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Uint32__Ouint32,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Assertions__Oassn_001t__Assertions__Oassn,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__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Uint32__Ouint32_001t__Uint32__Ouint32,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
    produc5703948589228662326_num_o: ( num > num > $o ) > product_prod_num_num > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    produc2098883955491630340Heap_o: ( num > num > heap_Time_Heap_o ) > product_prod_num_num > heap_Time_Heap_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    produc3422757380792381434ap_nat: ( num > num > heap_Time_Heap_nat ) > product_prod_num_num > heap_Time_Heap_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc1633297657014491743_VEBTi: ( num > num > heap_T8145700208782473153_VEBTi ) > product_prod_num_num > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Nat__Onat,type,
    produc2914010905598588082um_nat: ( num > num > nat ) > product_prod_num_num > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
    produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
    produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    produc1023323404773863986T_VEBT: ( num > num > set_VEBT_VEBT ) > product_prod_num_num > set_VEBT_VEBT ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    produc2624314226134418078_VEBTi: ( option4927543243414619207at_nat > produc8398139464844984134T_VEBT > heap_T8145700208782473153_VEBTi ) > produc819165548630102716T_VEBT > heap_T8145700208782473153_VEBTi ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_I_Eo_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Option__Ooption_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__VEBT____BuildupMemImp__OVEBTi_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_I_Eo_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Option__Ooption_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__VEBT____BuildupMemImp__OVEBTi_M_062_It__Nat__Onat_M_Eo_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8911080112929139129on_nat: ( vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____BuildupMemImp__OVEBTi_001t__Nat__Onat_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_I_Eo_M_062_It__Nat__Onat_M_Eo_J_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Option__Ooption_It__Nat__Onat_J_M_062_It__Nat__Onat_M_Eo_J_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_M_062_It__VEBT____BuildupMemImp__OVEBTi_M_062_It__Nat__Onat_M_Eo_J_J_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_I_Eo_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi_001_062_It__Nat__Onat_Mt__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
    produc758997459209783180T_VEBT: produc8398139464844984134T_VEBT > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_Eo,type,
    product_snd_nat_o: product_prod_nat_o > $o ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Int__Oint,type,
    product_snd_nat_int: product_prod_nat_int > int ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__VEBT____BuildupMemImp__OVEBTi,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Pure_Otype_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
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thf(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

thf(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

thf(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod_int_int ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: complex > real ).

thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: real > real ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: real > complex ).

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: real > real ).

thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
    refine_Imp_assert: $o > heap_T5738788834812785303t_unit ).

thf(sy_c_Refine__Imp__Hol_Orefines_001_Eo,type,
    refine_Imp_refines_o: heap_Time_Heap_o > heap_Time_Heap_o > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_I_Eo_J,type,
    refine5896690332125372649list_o: heap_T844314716496656296list_o > heap_T844314716496656296list_o > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    refine1935026298455697829on_nat: heap_T5317711798761887292on_nat > heap_T5317711798761887292on_nat > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    refine3700189196150522554_VEBTi: heap_T4980287057938770641_VEBTi > heap_T4980287057938770641_VEBTi > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__Nat__Onat,type,
    refine1365783493865988805es_nat: heap_Time_Heap_nat > heap_Time_Heap_nat > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__Option__Ooption_It__Nat__Onat_J,type,
    refine7594492741263601813on_nat: heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat > $o ).

thf(sy_c_Refine__Imp__Hol_Orefines_001t__VEBT____BuildupMemImp__OVEBTi,type,
    refine5565527176597971370_VEBTi: heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi > $o ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
    divide_divide_int: int > int > int ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
    divide_divide_nat: nat > nat > nat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > rat > rat ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
    divide_divide_real: real > real > real ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Uint32__Ouint32,type,
    divide_divide_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Rings_Odivide__class_Odivide_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    divide1791077408188789448l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: int > int > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: real > real > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Uint32__Ouint32,type,
    dvd_dvd_uint32: uint32 > uint32 > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    dvd_dv6812691276156420380l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > $o ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

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

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

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Uint32__Ouint32,type,
    modulo_modulo_uint32: uint32 > uint32 > uint32 ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    modulo1504961113040953224l_num1: word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: ( nat > complex ) > complex ).

thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
    suminf_int: ( nat > int ) > int ).

thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
    suminf_nat: ( nat > nat ) > nat ).

thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: ( nat > real ) > real ).

thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: ( nat > complex ) > $o ).

thf(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: ( nat > int ) > $o ).

thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
    summable_nat: ( nat > nat ) > $o ).

thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: ( nat > real ) > $o ).

thf(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: ( nat > real ) > real > $o ).

thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    collec1047362574656026267T_VEBT: ( produc8398139464844984134T_VEBT > $o ) > set_Pr563407847431865468T_VEBT ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Uint32__Ouint32,type,
    collect_uint32: ( uint32 > $o ) > set_uint32 ).

thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set_OCollect_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
    collec7814023847061821259l_num1: ( word_N3645301735248828278l_num1 > $o ) > set_wo3913738467083021356l_num1 ).

thf(sy_c_Set_Oimage_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    image_4470545334726330049nteger: ( code_integer > code_integer ) > set_Code_integer > set_Code_integer ).

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__Code____Numeral__Ointeger,type,
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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__String__Ochar,type,
    image_nat_char: ( nat > char ) > set_nat > set_char ).

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_Oimage_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    image_VEBT_VEBT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > set_nat ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__Code____Numeral__Ointeger,type,
    insert_Code_integer: code_integer > set_Code_integer > set_Code_integer ).

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__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    insert5290817439147925377n_assn: produc6575502325842934193n_assn > set_Pr5949110396991348497n_assn > set_Pr5949110396991348497n_assn ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    insert5033312907999012233nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

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__VEBT____BuildupMemImp__OVEBTi,type,
    insert_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > set_VEBT_VEBTi ).

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__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Code____Numeral__Ointeger,type,
    set_or8404916559141939852nteger: code_integer > code_integer > set_Code_integer ).

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,
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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_OgreaterThanLessThan_001t__Int__Oint,type,
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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_OlessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Int__Oint,type,
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thf(sy_c_Signed__Division_Osigned__division__class_Osigned__divide_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J,type,
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thf(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo_001t__Int__Oint,type,
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thf(sy_c_String_Oascii__of,type,
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thf(sy_c_String_Ochar_OChar,type,
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thf(sy_c_String_Ochar_Osize__char,type,
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thf(sy_c_String_Ochar__of__integer,type,
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thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
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thf(sy_c_String_Ointeger__of__char,type,
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thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001_Eo,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_I_Eo_J,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Int__Oint_J,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__Nat__Onat_J,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
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thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_I_Eo_J,type,
    time_TBOUND_list_o: heap_T844314716496656296list_o > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Nat__Onat_J,type,
    time_TBOUND_list_nat: heap_T290393402774840812st_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__Option__Ooption_It__Nat__Onat_J_J,type,
    time_T3808005469503390304on_nat: heap_T5317711798761887292on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_T8149879359713347829_VEBTi: heap_T4980287057938770641_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Nat__Onat,type,
    time_TBOUND_nat: heap_Time_Heap_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_T8353473612707095248on_nat: heap_T2636463487746394924on_nat > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__Product____Type__Ounit,type,
    time_T7469515765551943773t_unit: heap_T5738788834812785303t_unit > nat > $o ).

thf(sy_c_Time__Reasoning_OTBOUND_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_T5737551269749752165_VEBTi: heap_T8145700208782473153_VEBTi > nat > $o ).

thf(sy_c_Time__Reasoning_Ofails_001t__Product____Type__Ounit,type,
    time_f8834461667527620124t_unit: heap_T5738788834812785303t_unit > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001_Eo,type,
    time_htt_o: assn > heap_Time_Heap_o > ( $o > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Nat__Onat,type,
    time_htt_nat: assn > heap_Time_Heap_nat > ( nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_htt_option_nat: assn > heap_T2636463487746394924on_nat > ( option_nat > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Ohtt_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_htt_VEBT_VEBTi: assn > heap_T8145700208782473153_VEBTi > ( vEBT_VEBTi > assn ) > nat > $o ).

thf(sy_c_Time__Reasoning_Otime_001_Eo,type,
    time_time_o: heap_Time_Heap_o > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_I_Eo_J,type,
    time_time_array_o: heap_T5660665574680485309rray_o > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Int__Oint_J,type,
    time_time_array_int: heap_T1346037964561226099ay_int > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__Nat__Onat_J,type,
    time_time_array_nat: heap_T3836121109492952855ay_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Heap__Oarray_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t9122064381910598399_VEBTi: heap_T8822477325091257596_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__Product____Type__Ounit_J,type,
    time_t4781937132199089312t_unit: heap_T7268547540234007069t_unit > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    time_t3534373299052942712_VEBTi: heap_T4980287057938770641_VEBTi > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Nat__Onat,type,
    time_time_nat: heap_Time_Heap_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Option__Ooption_It__Nat__Onat_J,type,
    time_time_option_nat: heap_T2636463487746394924on_nat > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__Product____Type__Ounit,type,
    time_t4224138285095624986t_unit: heap_T5738788834812785303t_unit > heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Time__Reasoning_Otime_001t__VEBT____BuildupMemImp__OVEBTi,type,
    time_time_VEBT_VEBTi: heap_T8145700208782473153_VEBTi > heap_e7401611519738050253t_unit > nat ).

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_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_Transcendental_Oarccos,type,
    arccos: real > real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

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

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____1,type,
    type_l31302759751748491nite_1: itself_finite_1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____2,type,
    type_l31302759751748492nite_2: itself_finite_2 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Enum__Ofinite____3,type,
    type_l31302759751748493nite_3: itself_finite_3 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    type_l796852477590012082l_num1: itself8794530163899892676l_num1 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum0,type,
    type_l4264026598287037464l_num0: itself_Numeral_num0 > nat ).

thf(sy_c_Type__Length_Olen0__class_Olen__of_001t__Numeral____Type__Onum1,type,
    type_l4264026598287037465l_num1: itself_Numeral_num1 > nat ).

thf(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > word_N3645301735248828278l_num1 ).

thf(sy_c_Uint32_OUint32,type,
    uint322: code_integer > uint32 ).

thf(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

thf(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: uint32 > code_integer ).

thf(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

thf(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

thf(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( nat > $o ) > nat > uint32 > uint32 ).

thf(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: nat > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: word_N3645301735248828278l_num1 > uint32 ).

thf(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word_N3645301735248828278l_num1 ).

thf(sy_c_Uint32_Ouint32__div,type,
    uint32_div: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: uint32 > uint32 > produc827990862158126777uint32 ).

thf(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: uint32 > uint32 > uint32 ).

thf(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: uint32 > code_integer > $o > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: uint32 > code_integer > uint32 ).

thf(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: uint32 > code_integer > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: nat > nat > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001_Eo,type,
    vEBT_V2326993469660664182atei_o: nat > heap_Time_Heap_o > heap_T844314716496656296list_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Nat__Onat,type,
    vEBT_V7726092123322077554ei_nat: nat > heap_Time_Heap_nat > heap_T290393402774840812st_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Option__Ooption_It__Nat__Onat_J,type,
    vEBT_V792416675989592002on_nat: nat > heap_T2636463487746394924on_nat > heap_T5317711798761887292on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__Product____Type__Ounit,type,
    vEBT_V7483891112628345579t_unit: nat > heap_T5738788834812785303t_unit > heap_T7268547540234007069t_unit ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_V1859673955506687831_VEBTi: nat > heap_T8145700208782473153_VEBTi > heap_T4980287057938770641_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H__rel,type,
    vEBT_V254170901696579886pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: $o > $o > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > vEBT_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_I_Eo_J,type,
    vEBT_c6104975476656191286Heap_o: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o ) > ( $o > $o > heap_Time_Heap_o ) > vEBT_VEBTi > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J,type,
    vEBT_c1335663792808957512ap_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat ) > ( $o > $o > heap_Time_Heap_nat ) > vEBT_VEBTi > heap_Time_Heap_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J,type,
    vEBT_c6250501799366334488on_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat ) > ( $o > $o > heap_T2636463487746394924on_nat ) > vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J,type,
    vEBT_c6028912655521741485_VEBTi: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ) > ( $o > $o > heap_T8145700208782473153_VEBTi ) > vEBT_VEBTi > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi_001t__Nat__Onat,type,
    vEBT_case_VEBTi_nat: ( option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat ) > ( $o > $o > nat ) > vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: vEBT_VEBTi > nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: vEBT_VEBT > vEBT_VEBTi > assn ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi__rel,type,
    vEBT_v1230518104690509829pi_rel: nat > nat > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: vEBT_VEBTi > nat > heap_Time_Heap_o ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: vEBT_VEBTi > vEBT_VEBTi > $o ).

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

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

thf(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001_Eo,type,
    vEBT_case_VEBT_o: ( option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o ) > ( $o > $o > $o ) > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J_J,type,
    vEBT_c634343235235684882T_VEBT: ( option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT ) > ( $o > $o > produc819165548630102716T_VEBT ) > vEBT_VEBT > produc819165548630102716T_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT_Ois__Node,type,
    vEBT_is_Node: vEBT_VEBT > $o ).

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__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: vEBT_VEBT > nat > nat ).

thf(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_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__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: vEBT_VEBT > nat ).

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

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

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

thf(sy_c_VEBT__List__Assn_OlistI__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L1528199826722428489_VEBTi: set_nat > ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

thf(sy_c_VEBT__List__Assn_Olist__assn_001t__VEBT____Definitions__OVEBT_001t__VEBT____BuildupMemImp__OVEBTi,type,
    vEBT_L6296928887356842470_VEBTi: ( vEBT_VEBT > vEBT_VEBTi > assn ) > list_VEBT_VEBT > list_VEBT_VEBTi > assn ).

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_I_Eo_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V613753007643960916it_nat: ( produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat ) > option7339022715339332451it_nat > option7339022715339332451it_nat > option7339022715339332451it_nat ).

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_Ooption__shift_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Nat__Onat_J_J,type,
    vEBT_V819568868292977612it_nat: ( produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat ) > option2621746655072343315it_nat > option2621746655072343315it_nat > option2621746655072343315it_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__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: nat > nat > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: vEBT_VEBT > real ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: vEBT_VEBT > nat ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: vEBT_VEBT > vEBT_VEBT > $o ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    accp_P7675410724331315407_VEBTi: ( produc3625547720036274456_VEBTi > produc3625547720036274456_VEBTi > $o ) > produc3625547720036274456_VEBTi > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__VEBT____BuildupMemImp__OVEBTi,type,
    accp_VEBT_VEBTi: ( vEBT_VEBTi > vEBT_VEBTi > $o ) > vEBT_VEBTi > $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_Word_Oeven__word_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    even_w9054469088133485505l_num1: word_N3645301735248828278l_num1 > $o ).

thf(sy_c_Word_Osemiring__1__class_Ounsigned_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_001t__Int__Oint,type,
    semiri7338730514057886004m1_int: word_N3645301735248828278l_num1 > int ).

thf(sy_c_Word_Osigned__drop__bit_001t__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J,type,
    signed5000768011106662067l_num1: nat > word_N3645301735248828278l_num1 > word_N3645301735248828278l_num1 ).

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__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $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__Real__Oreal_J,type,
    member_list_real: list_real > set_list_real > $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__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    member7957490590177025114n_assn: produc6575502325842934193n_assn > set_Pr5949110396991348497n_assn > $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__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
    member6310962623043647828_nat_o: product_prod_nat_o > set_Pr3149072824959771635_nat_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
    member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__VEBT____Definitions__OVEBT_J_J,type,
    member306291179834725981T_VEBT: produc8398139464844984134T_VEBT > set_Pr563407847431865468T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    member557208447399453958t_real: produc7716430852924023517t_real > set_Pr320017278500174781t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____BuildupMemImp__OVEBTi_J,type,
    member763447850065367567_VEBTi: produc214224863196444774_VEBTi > set_Pr1938536134445252166_VEBTi > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: produc8025551001238799321T_VEBT > set_Pr6167073792073659919T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).

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

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

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

thf(sy_c_member_001t__VEBT____BuildupMemImp__OVEBTi,type,
    member_VEBT_VEBTi: vEBT_VEBTi > set_VEBT_VEBTi > $o ).

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

thf(sy_v_f____,type,
    f: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ).

thf(sy_v_ta____,type,
    ta: vEBT_VEBT ).

thf(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

thf(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (10176)
thf(fact_0_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T2636463487746394924on_nat,F12: $o > $o > heap_T2636463487746394924on_nat,F2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat] :
      ( ( Ti = Ti2 )
     => ( ! [A: $o,B: $o] : ( refine7594492741263601813on_nat @ ( F1 @ A @ B ) @ ( F12 @ A @ B ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : ( refine7594492741263601813on_nat @ ( F2 @ Info @ Deg @ TreeArray @ Summary ) @ ( F22 @ Info @ Deg @ TreeArray @ Summary ) )
         => ( refine7594492741263601813on_nat @ ( vEBT_c6250501799366334488on_nat @ F2 @ F1 @ Ti ) @ ( vEBT_c6250501799366334488on_nat @ F22 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_1_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_Time_Heap_o,F12: $o > $o > heap_Time_Heap_o,F2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( Ti = Ti2 )
     => ( ! [A: $o,B: $o] : ( refine_Imp_refines_o @ ( F1 @ A @ B ) @ ( F12 @ A @ B ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F2 @ Info @ Deg @ TreeArray @ Summary ) @ ( F22 @ Info @ Deg @ TreeArray @ Summary ) )
         => ( refine_Imp_refines_o @ ( vEBT_c6104975476656191286Heap_o @ F2 @ F1 @ Ti ) @ ( vEBT_c6104975476656191286Heap_o @ F22 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_2_refines__case__VEBTi,axiom,
    ! [Ti: vEBT_VEBTi,Ti2: vEBT_VEBTi,F1: $o > $o > heap_T8145700208782473153_VEBTi,F12: $o > $o > heap_T8145700208782473153_VEBTi,F2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F22: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( Ti = Ti2 )
     => ( ! [A: $o,B: $o] : ( refine5565527176597971370_VEBTi @ ( F1 @ A @ B ) @ ( F12 @ A @ B ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F2 @ Info @ Deg @ TreeArray @ Summary ) @ ( F22 @ Info @ Deg @ TreeArray @ Summary ) )
         => ( refine5565527176597971370_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F2 @ F1 @ Ti ) @ ( vEBT_c6028912655521741485_VEBTi @ F22 @ F12 @ Ti2 ) ) ) ) ) ).

% refines_case_VEBTi
thf(fact_3_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_4_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_5_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_6_power__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( power_power_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% power_shift
thf(fact_7_bit__split__inv,axiom,
    ! [X2: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
      = X2 ) ).

% bit_split_inv
thf(fact_8_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_9_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_10_vebt__predi_Osimps,axiom,
    ( vEBT_vebt_predi
    = ( ^ [T: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ^ [Mima: product_prod_nat_nat] :
                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                      @ ^ [L: nat] :
                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                          @ ^ [H: nat] :
                              ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                              @ ^ [Aktnode: vEBT_VEBTi] :
                                  ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                  @ ^ [Minlow: option_nat] :
                                      ( if_Hea5867803462524415986on_nat
                                      @ ( ( Minlow != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Aktnode @ L )
                                        @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                      @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Summary2 @ H )
                                        @ ^ [Predsum: option_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                  ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                  @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info2 )
          @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
          @ T ) ) ) ).

% vebt_predi.simps
thf(fact_11_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_predi
    = ( ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
              @ ^ [Uu: product_unit] :
                  ( produc2190226783763740553on_nat
                  @ ^ [Info3: option4927543243414619207at_nat] :
                      ( produc2606485630176857543on_nat
                      @ ^ [Deg3: nat] :
                          ( produc6124225815318652422on_nat
                          @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                              ( heap_T5999496708990702694on_nat
                              @ ( refine_Imp_assert
                                @ ( ( Info3 = Info2 )
                                  & ( Deg3 = Deg2 )
                                  & ( vEBT_is_Node @ T ) ) )
                              @ ^ [Uv: product_unit] :
                                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                  @ ^ [Mima: product_prod_nat_nat] :
                                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                                        @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                          @ ^ [L: nat] :
                                              ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                              @ ^ [H: nat] :
                                                  ( heap_T5999496708990702694on_nat
                                                  @ ( refine_Imp_assert
                                                    @ ( L
                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ^ [Uw: product_unit] :
                                                      ( heap_T5999496708990702694on_nat
                                                      @ ( refine_Imp_assert
                                                        @ ( H
                                                          = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      @ ^ [Ux: product_unit] :
                                                          ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                          @ ^ [Uy: product_unit] :
                                                              ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                              @ ^ [Aktnode: vEBT_VEBTi] :
                                                                  ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                                                  @ ^ [Minlow: option_nat] :
                                                                      ( heap_T5999496708990702694on_nat
                                                                      @ ( refine_Imp_assert
                                                                        @ ( Minlow
                                                                          = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                      @ ^ [Uz: product_unit] :
                                                                          ( if_Hea5867803462524415986on_nat
                                                                          @ ( ( Minlow != none_nat )
                                                                            & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                                          @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                            @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                          @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_predi @ Summary3 @ Summary2 @ H )
                                                                            @ ^ [Predsum: option_nat] :
                                                                                ( heap_T5999496708990702694on_nat
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( ( Predsum = none_nat )
                                                                                    = ( ( vEBT_vebt_pred @ Summary3 @ H )
                                                                                      = none_nat ) ) )
                                                                                @ ^ [Va: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                    @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                                                      @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                          ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                          @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info2 ) ) ) )
                  @ ( vEBT_c634343235235684882T_VEBT
                    @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                    @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                    @ T ) ) )
          @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_predi'.simps
thf(fact_12_refines__replicate,axiom,
    ! [F: heap_T2636463487746394924on_nat,F3: heap_T2636463487746394924on_nat,N2: nat] :
      ( ( refine7594492741263601813on_nat @ F @ F3 )
     => ( refine1935026298455697829on_nat @ ( vEBT_V792416675989592002on_nat @ N2 @ F ) @ ( vEBT_V792416675989592002on_nat @ N2 @ F3 ) ) ) ).

% refines_replicate
thf(fact_13_refines__replicate,axiom,
    ! [F: heap_Time_Heap_o,F3: heap_Time_Heap_o,N2: nat] :
      ( ( refine_Imp_refines_o @ F @ F3 )
     => ( refine5896690332125372649list_o @ ( vEBT_V2326993469660664182atei_o @ N2 @ F ) @ ( vEBT_V2326993469660664182atei_o @ N2 @ F3 ) ) ) ).

% refines_replicate
thf(fact_14_refines__replicate,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,F3: heap_T8145700208782473153_VEBTi,N2: nat] :
      ( ( refine5565527176597971370_VEBTi @ F @ F3 )
     => ( refine3700189196150522554_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ F ) @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ F3 ) ) ) ).

% refines_replicate
thf(fact_15_power__decreasing__iff,axiom,
    ! [B3: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ B3 @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B3 @ M ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_16_power__decreasing__iff,axiom,
    ! [B3: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( ord_less_real @ B3 @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_17_power__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B3 )
     => ( ( ord_less_rat @ B3 @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_18_power__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ B3 @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_19_power__decreasing__iff,axiom,
    ! [B3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ B3 @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_20_zero__less__power2,axiom,
    ! [A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A3 != zero_z3403309356797280102nteger ) ) ).

% zero_less_power2
thf(fact_21_zero__less__power2,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A3 != zero_zero_real ) ) ).

% zero_less_power2
thf(fact_22_zero__less__power2,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A3 != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_23_zero__less__power2,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A3 != zero_zero_int ) ) ).

% zero_less_power2
thf(fact_24_power2__less__eq__zero__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% power2_less_eq_zero_iff
thf(fact_25_power2__less__eq__zero__iff,axiom,
    ! [A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% power2_less_eq_zero_iff
thf(fact_26_power2__less__eq__zero__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_27_power2__less__eq__zero__iff,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% power2_less_eq_zero_iff
thf(fact_28_power2__eq__iff__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_29_power2__eq__iff__nonneg,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
       => ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_30_power2__eq__iff__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_31_power2__eq__iff__nonneg,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_32_power2__eq__iff__nonneg,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X2 = Y2 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_33_bits__1__div__2,axiom,
    ( ( divide1791077408188789448l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% bits_1_div_2
thf(fact_34_bits__1__div__2,axiom,
    ( ( divide_divide_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = zero_zero_uint32 ) ).

% bits_1_div_2
thf(fact_35_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_36_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_37_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_38_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_39_power__increasing__iff,axiom,
    ! [B3: code_integer,X2: nat,Y2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B3 @ X2 ) @ ( power_8256067586552552935nteger @ B3 @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_40_power__increasing__iff,axiom,
    ! [B3: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_41_power__increasing__iff,axiom,
    ! [B3: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_42_power__increasing__iff,axiom,
    ! [B3: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B3 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_43_power__increasing__iff,axiom,
    ! [B3: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y2 ) )
        = ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power_increasing_iff
thf(fact_44_power__mono__iff,axiom,
    ! [A3: real,B3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) )
            = ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_45_power__mono__iff,axiom,
    ! [A3: code_integer,B3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) )
            = ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_46_power__mono__iff,axiom,
    ! [A3: rat,B3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) )
            = ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_47_power__mono__iff,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) )
            = ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_48_power__mono__iff,axiom,
    ! [A3: int,B3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) )
            = ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ) ).

% power_mono_iff
thf(fact_49_power__strict__decreasing__iff,axiom,
    ! [B3: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ B3 @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B3 @ M ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_50_power__strict__decreasing__iff,axiom,
    ! [B3: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( ord_less_real @ B3 @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_51_power__strict__decreasing__iff,axiom,
    ! [B3: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B3 )
     => ( ( ord_less_rat @ B3 @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_52_power__strict__decreasing__iff,axiom,
    ! [B3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ B3 @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_53_power__strict__decreasing__iff,axiom,
    ! [B3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ B3 @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_54_zero__eq__power2,axiom,
    ! [A3: rat] :
      ( ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_55_zero__eq__power2,axiom,
    ! [A3: nat] :
      ( ( ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% zero_eq_power2
thf(fact_56_zero__eq__power2,axiom,
    ! [A3: int] :
      ( ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% zero_eq_power2
thf(fact_57_zero__eq__power2,axiom,
    ! [A3: real] :
      ( ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% zero_eq_power2
thf(fact_58_zero__eq__power2,axiom,
    ! [A3: complex] :
      ( ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_complex )
      = ( A3 = zero_zero_complex ) ) ).

% zero_eq_power2
thf(fact_59_zero__eq__power2,axiom,
    ! [A3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% zero_eq_power2
thf(fact_60_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_61_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_62_bits__div__by__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_by_0
thf(fact_63_bits__div__by__0,axiom,
    ! [A3: uint32] :
      ( ( divide_divide_uint32 @ A3 @ zero_zero_uint32 )
      = zero_zero_uint32 ) ).

% bits_div_by_0
thf(fact_64_bits__div__by__0,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ A3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_65_bits__div__by__0,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ A3 @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_66_bits__div__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_div_0
thf(fact_67_bits__div__0,axiom,
    ! [A3: uint32] :
      ( ( divide_divide_uint32 @ zero_zero_uint32 @ A3 )
      = zero_zero_uint32 ) ).

% bits_div_0
thf(fact_68_bits__div__0,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_69_bits__div__0,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ zero_zero_int @ A3 )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_70_bits__div__by__1,axiom,
    ! [A3: uint32] :
      ( ( divide_divide_uint32 @ A3 @ one_one_uint32 )
      = A3 ) ).

% bits_div_by_1
thf(fact_71_bits__div__by__1,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ A3 @ one_one_nat )
      = A3 ) ).

% bits_div_by_1
thf(fact_72_bits__div__by__1,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ A3 @ one_one_int )
      = A3 ) ).

% bits_div_by_1
thf(fact_73_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_uint32 @ one_one_uint32 @ N2 )
      = one_one_uint32 ) ).

% power_one
thf(fact_74_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_75_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_76_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_77_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_78_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ one_one_complex @ N2 )
      = one_one_complex ) ).

% power_one
thf(fact_79_power__one,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ one_one_Code_integer @ N2 )
      = one_one_Code_integer ) ).

% power_one
thf(fact_80_power__one__right,axiom,
    ! [A3: nat] :
      ( ( power_power_nat @ A3 @ one_one_nat )
      = A3 ) ).

% power_one_right
thf(fact_81_power__one__right,axiom,
    ! [A3: int] :
      ( ( power_power_int @ A3 @ one_one_nat )
      = A3 ) ).

% power_one_right
thf(fact_82_power__one__right,axiom,
    ! [A3: real] :
      ( ( power_power_real @ A3 @ one_one_nat )
      = A3 ) ).

% power_one_right
thf(fact_83_power__one__right,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ A3 @ one_one_nat )
      = A3 ) ).

% power_one_right
thf(fact_84_power__one__right,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ A3 @ one_one_nat )
      = A3 ) ).

% power_one_right
thf(fact_85_power__inject__exp,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ( ( power_8256067586552552935nteger @ A3 @ M )
          = ( power_8256067586552552935nteger @ A3 @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_86_power__inject__exp,axiom,
    ! [A3: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ( power_power_real @ A3 @ M )
          = ( power_power_real @ A3 @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_87_power__inject__exp,axiom,
    ! [A3: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ( ( power_power_rat @ A3 @ M )
          = ( power_power_rat @ A3 @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_88_power__inject__exp,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ( ( power_power_nat @ A3 @ M )
          = ( power_power_nat @ A3 @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_89_power__inject__exp,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ( ( power_power_int @ A3 @ M )
          = ( power_power_int @ A3 @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_90_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ K ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_zero_numeral
thf(fact_91_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_92_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_93_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_94_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_95_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_96_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ K ) )
      = zero_z3403309356797280102nteger ) ).

% power_zero_numeral
thf(fact_97_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_98_nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_99_power__strict__increasing__iff,axiom,
    ! [B3: code_integer,X2: nat,Y2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B3 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B3 @ X2 ) @ ( power_8256067586552552935nteger @ B3 @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_100_power__strict__increasing__iff,axiom,
    ! [B3: real,X2: nat,Y2: nat] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_101_power__strict__increasing__iff,axiom,
    ! [B3: rat,X2: nat,Y2: nat] :
      ( ( ord_less_rat @ one_one_rat @ B3 )
     => ( ( ord_less_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_102_power__strict__increasing__iff,axiom,
    ! [B3: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ one_one_nat @ B3 )
     => ( ( ord_less_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_103_power__strict__increasing__iff,axiom,
    ! [B3: int,X2: nat,Y2: nat] :
      ( ( ord_less_int @ one_one_int @ B3 )
     => ( ( ord_less_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y2 ) )
        = ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power_strict_increasing_iff
thf(fact_104_power__eq__0__iff,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ( power_power_rat @ A3 @ N2 )
        = zero_zero_rat )
      = ( ( A3 = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_105_power__eq__0__iff,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ( power_power_nat @ A3 @ N2 )
        = zero_zero_nat )
      = ( ( A3 = zero_zero_nat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_106_power__eq__0__iff,axiom,
    ! [A3: int,N2: nat] :
      ( ( ( power_power_int @ A3 @ N2 )
        = zero_zero_int )
      = ( ( A3 = zero_zero_int )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_107_power__eq__0__iff,axiom,
    ! [A3: real,N2: nat] :
      ( ( ( power_power_real @ A3 @ N2 )
        = zero_zero_real )
      = ( ( A3 = zero_zero_real )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_108_power__eq__0__iff,axiom,
    ! [A3: complex,N2: nat] :
      ( ( ( power_power_complex @ A3 @ N2 )
        = zero_zero_complex )
      = ( ( A3 = zero_zero_complex )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_109_power__eq__0__iff,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ A3 @ N2 )
        = zero_z3403309356797280102nteger )
      = ( ( A3 = zero_z3403309356797280102nteger )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_110_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_111_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_112_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_113_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: rat,N2: nat] :
      ( ( A3 != zero_zero_rat )
     => ( ( power_power_rat @ A3 @ N2 )
       != zero_zero_rat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_114_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: nat,N2: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( power_power_nat @ A3 @ N2 )
       != zero_zero_nat ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_115_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: int,N2: nat] :
      ( ( A3 != zero_zero_int )
     => ( ( power_power_int @ A3 @ N2 )
       != zero_zero_int ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_116_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: real,N2: nat] :
      ( ( A3 != zero_zero_real )
     => ( ( power_power_real @ A3 @ N2 )
       != zero_zero_real ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_117_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: complex,N2: nat] :
      ( ( A3 != zero_zero_complex )
     => ( ( power_power_complex @ A3 @ N2 )
       != zero_zero_complex ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_118_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ A3 @ N2 )
       != zero_z3403309356797280102nteger ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
thf(fact_119_power__divide,axiom,
    ! [A3: complex,B3: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A3 @ B3 ) @ N2 )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A3 @ N2 ) @ ( power_power_complex @ B3 @ N2 ) ) ) ).

% power_divide
thf(fact_120_power__divide,axiom,
    ! [A3: real,B3: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A3 @ B3 ) @ N2 )
      = ( divide_divide_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ).

% power_divide
thf(fact_121_power__divide,axiom,
    ! [A3: rat,B3: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A3 @ B3 ) @ N2 )
      = ( divide_divide_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ).

% power_divide
thf(fact_122_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_123_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_124_zero__le__power,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% zero_le_power
thf(fact_125_zero__le__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% zero_le_power
thf(fact_126_zero__le__power,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% zero_le_power
thf(fact_127_zero__le__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% zero_le_power
thf(fact_128_zero__le__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) ) ) ).

% zero_le_power
thf(fact_129_power__mono,axiom,
    ! [A3: real,B3: real,N2: nat] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ) ).

% power_mono
thf(fact_130_power__mono,axiom,
    ! [A3: code_integer,B3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ) ).

% power_mono
thf(fact_131_power__mono,axiom,
    ! [A3: rat,B3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ) ).

% power_mono
thf(fact_132_power__mono,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ) ).

% power_mono
thf(fact_133_power__mono,axiom,
    ! [A3: int,B3: int,N2: nat] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ) ).

% power_mono
thf(fact_134_zero__less__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% zero_less_power
thf(fact_135_zero__less__power,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% zero_less_power
thf(fact_136_zero__less__power,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% zero_less_power
thf(fact_137_zero__less__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% zero_less_power
thf(fact_138_zero__less__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) ) ) ).

% zero_less_power
thf(fact_139_one__le__power,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A3 )
     => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% one_le_power
thf(fact_140_one__le__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A3 )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% one_le_power
thf(fact_141_one__le__power,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A3 )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% one_le_power
thf(fact_142_one__le__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A3 )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% one_le_power
thf(fact_143_one__le__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A3 )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A3 @ N2 ) ) ) ).

% one_le_power
thf(fact_144_mem__Collect__eq,axiom,
    ! [A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( member_VEBT_VEBT @ A3 @ ( collect_VEBT_VEBT @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_145_mem__Collect__eq,axiom,
    ! [A3: real,P: real > $o] :
      ( ( member_real @ A3 @ ( collect_real @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( member_nat @ A3 @ ( collect_nat @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
    ! [A3: int,P: int > $o] :
      ( ( member_int @ A3 @ ( collect_int @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
    ! [A3: complex,P: complex > $o] :
      ( ( member_complex @ A3 @ ( collect_complex @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
    ! [A3: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( member5262025264175285858nt_int @ A3 @ ( collec213857154873943460nt_int @ P ) )
      = ( P @ A3 ) ) ).

% mem_Collect_eq
thf(fact_150_Collect__mem__eq,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_155_Collect__mem__eq,axiom,
    ! [A4: set_Pr958786334691620121nt_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_156_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_157_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_158_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_159_Collect__cong,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ! [X3: product_prod_int_int] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec213857154873943460nt_int @ P )
        = ( collec213857154873943460nt_int @ Q ) ) ) ).

% Collect_cong
thf(fact_160_power__one__over,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A3 ) @ N2 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_one_over
thf(fact_161_power__one__over,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A3 ) @ N2 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% power_one_over
thf(fact_162_power__one__over,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_one_over
thf(fact_163_power__0,axiom,
    ! [A3: uint32] :
      ( ( power_power_uint32 @ A3 @ zero_zero_nat )
      = one_one_uint32 ) ).

% power_0
thf(fact_164_power__0,axiom,
    ! [A3: rat] :
      ( ( power_power_rat @ A3 @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_165_power__0,axiom,
    ! [A3: nat] :
      ( ( power_power_nat @ A3 @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_166_power__0,axiom,
    ! [A3: int] :
      ( ( power_power_int @ A3 @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_167_power__0,axiom,
    ! [A3: real] :
      ( ( power_power_real @ A3 @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_168_power__0,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ A3 @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_169_power__0,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ A3 @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_170_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_171_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_172_power__less__imp__less__base,axiom,
    ! [A3: code_integer,N2: nat,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ord_le6747313008572928689nteger @ A3 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_173_power__less__imp__less__base,axiom,
    ! [A3: real,N2: nat,B3: real] :
      ( ( ord_less_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ A3 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_174_power__less__imp__less__base,axiom,
    ! [A3: rat,N2: nat,B3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ A3 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_175_power__less__imp__less__base,axiom,
    ! [A3: nat,N2: nat,B3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ A3 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_176_power__less__imp__less__base,axiom,
    ! [A3: int,N2: nat,B3: int] :
      ( ( ord_less_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ A3 @ B3 ) ) ) ).

% power_less_imp_less_base
thf(fact_177_power__le__one,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ A3 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ one_one_real ) ) ) ).

% power_le_one
thf(fact_178_power__le__one,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le3102999989581377725nteger @ A3 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_179_power__le__one,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ A3 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_180_power__le__one,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N2 ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_181_power__le__one,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ A3 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_182_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_183_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_184_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N2 )
          = one_one_uint32 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_uint32 @ zero_zero_uint32 @ N2 )
          = zero_zero_uint32 ) ) ) ).

% power_0_left
thf(fact_185_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
          = one_on7727431528512463931l_num1 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% power_0_left
thf(fact_186_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_187_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = one_one_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N2 )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_188_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = one_one_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N2 )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_189_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = one_one_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_real @ zero_zero_real @ N2 )
          = zero_zero_real ) ) ) ).

% power_0_left
thf(fact_190_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = one_one_complex ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_complex @ zero_zero_complex @ N2 )
          = zero_zero_complex ) ) ) ).

% power_0_left
thf(fact_191_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
          = one_one_Code_integer ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_192_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A3: code_integer] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ A3 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_193_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A3: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ one_one_real @ A3 )
       => ( ord_less_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ A3 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_194_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A3: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ one_one_rat @ A3 )
       => ( ord_less_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ A3 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_195_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A3: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ one_one_nat @ A3 )
       => ( ord_less_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ A3 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_196_power__strict__increasing,axiom,
    ! [N2: nat,N3: nat,A3: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ one_one_int @ A3 )
       => ( ord_less_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ A3 @ N3 ) ) ) ) ).

% power_strict_increasing
thf(fact_197_power__less__imp__less__exp,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_198_power__less__imp__less__exp,axiom,
    ! [A3: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_199_power__less__imp__less__exp,axiom,
    ! [A3: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ( ord_less_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_200_power__less__imp__less__exp,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ( ord_less_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_201_power__less__imp__less__exp,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ( ord_less_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_202_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
        = zero_z3563351764282998399l_num1 ) ) ).

% zero_power
thf(fact_203_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_rat @ zero_zero_rat @ N2 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_204_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_nat @ zero_zero_nat @ N2 )
        = zero_zero_nat ) ) ).

% zero_power
thf(fact_205_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_int @ zero_zero_int @ N2 )
        = zero_zero_int ) ) ).

% zero_power
thf(fact_206_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ zero_zero_real @ N2 )
        = zero_zero_real ) ) ).

% zero_power
thf(fact_207_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ zero_zero_complex @ N2 )
        = zero_zero_complex ) ) ).

% zero_power
thf(fact_208_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N2 )
        = zero_z3403309356797280102nteger ) ) ).

% zero_power
thf(fact_209_power__increasing,axiom,
    ! [N2: nat,N3: nat,A3: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ one_one_real @ A3 )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ A3 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_210_power__increasing,axiom,
    ! [N2: nat,N3: nat,A3: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A3 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ A3 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_211_power__increasing,axiom,
    ! [N2: nat,N3: nat,A3: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A3 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ A3 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_212_power__increasing,axiom,
    ! [N2: nat,N3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A3 )
       => ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ A3 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_213_power__increasing,axiom,
    ! [N2: nat,N3: nat,A3: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ one_one_int @ A3 )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ A3 @ N3 ) ) ) ) ).

% power_increasing
thf(fact_214_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_215_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_216_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_217_zero__power2,axiom,
    ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3563351764282998399l_num1 ) ).

% zero_power2
thf(fact_218_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_219_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_220_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_221_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_222_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_223_zero__power2,axiom,
    ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% zero_power2
thf(fact_224_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: code_integer] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_le6747313008572928689nteger @ A3 @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N3 ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_225_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: real] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ( ord_less_real @ A3 @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A3 @ N3 ) @ ( power_power_real @ A3 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_226_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: rat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_rat @ A3 @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A3 @ N3 ) @ ( power_power_rat @ A3 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_227_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: nat] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_nat @ A3 @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A3 @ N3 ) @ ( power_power_nat @ A3 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_228_power__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: int] :
      ( ( ord_less_nat @ N2 @ N3 )
     => ( ( ord_less_int @ zero_zero_int @ A3 )
       => ( ( ord_less_int @ A3 @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A3 @ N3 ) @ ( power_power_int @ A3 @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_229_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: real] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ( ord_less_eq_real @ A3 @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A3 @ N3 ) @ ( power_power_real @ A3 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_230_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: code_integer] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_le3102999989581377725nteger @ A3 @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N3 ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_231_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: rat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_eq_rat @ A3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N3 ) @ ( power_power_rat @ A3 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_232_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_eq_nat @ A3 @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A3 @ N3 ) @ ( power_power_nat @ A3 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_233_power__decreasing,axiom,
    ! [N2: nat,N3: nat,A3: int] :
      ( ( ord_less_eq_nat @ N2 @ N3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ( ord_less_eq_int @ A3 @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A3 @ N3 ) @ ( power_power_int @ A3 @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_234_power__eq__imp__eq__base,axiom,
    ! [A3: real,N2: nat,B3: real] :
      ( ( ( power_power_real @ A3 @ N2 )
        = ( power_power_real @ B3 @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A3 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_235_power__eq__imp__eq__base,axiom,
    ! [A3: code_integer,N2: nat,B3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A3 @ N2 )
        = ( power_8256067586552552935nteger @ B3 @ N2 ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A3 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_236_power__eq__imp__eq__base,axiom,
    ! [A3: rat,N2: nat,B3: rat] :
      ( ( ( power_power_rat @ A3 @ N2 )
        = ( power_power_rat @ B3 @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A3 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_237_power__eq__imp__eq__base,axiom,
    ! [A3: nat,N2: nat,B3: nat] :
      ( ( ( power_power_nat @ A3 @ N2 )
        = ( power_power_nat @ B3 @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A3 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_238_power__eq__imp__eq__base,axiom,
    ! [A3: int,N2: nat,B3: int] :
      ( ( ( power_power_int @ A3 @ N2 )
        = ( power_power_int @ B3 @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A3 = B3 ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_239_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A3: real,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ( power_power_real @ A3 @ N2 )
              = ( power_power_real @ B3 @ N2 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_240_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A3: code_integer,B3: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
         => ( ( ( power_8256067586552552935nteger @ A3 @ N2 )
              = ( power_8256067586552552935nteger @ B3 @ N2 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_241_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A3: rat,B3: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ( power_power_rat @ A3 @ N2 )
              = ( power_power_rat @ B3 @ N2 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_242_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ( power_power_nat @ A3 @ N2 )
              = ( power_power_nat @ B3 @ N2 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_243_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A3: int,B3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ( power_power_int @ A3 @ N2 )
              = ( power_power_int @ B3 @ N2 ) )
            = ( A3 = B3 ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_244_one__power2,axiom,
    ( ( power_power_uint32 @ one_one_uint32 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% one_power2
thf(fact_245_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_246_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_247_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_248_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_249_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_250_one__power2,axiom,
    ( ( power_8256067586552552935nteger @ one_one_Code_integer @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_power2
thf(fact_251_power__le__imp__le__exp,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_252_power__le__imp__le__exp,axiom,
    ! [A3: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_253_power__le__imp__le__exp,axiom,
    ! [A3: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_254_power__le__imp__le__exp,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_255_power__le__imp__le__exp,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_256_self__le__power,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% self_le_power
thf(fact_257_self__le__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le3102999989581377725nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% self_le_power
thf(fact_258_self__le__power,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% self_le_power
thf(fact_259_self__le__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) ) ) ).

% self_le_power
thf(fact_260_self__le__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% self_le_power
thf(fact_261_one__less__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% one_less_power
thf(fact_262_one__less__power,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_real @ one_one_real @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% one_less_power
thf(fact_263_one__less__power,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% one_less_power
thf(fact_264_one__less__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ N2 ) ) ) ) ).

% one_less_power
thf(fact_265_one__less__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% one_less_power
thf(fact_266_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_267_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_268_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_269_power2__le__imp__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_270_power2__le__imp__le,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
       => ( ord_le3102999989581377725nteger @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_271_power2__le__imp__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_272_power2__le__imp__le,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_273_power2__le__imp__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% power2_le_imp_le
thf(fact_274_power2__eq__imp__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_275_power2__eq__imp__eq,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_276_power2__eq__imp__eq,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_277_power2__eq__imp__eq,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_278_power2__eq__imp__eq,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
         => ( X2 = Y2 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_279_zero__le__power2,axiom,
    ! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_280_zero__le__power2,axiom,
    ! [A3: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_281_zero__le__power2,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_282_zero__le__power2,axiom,
    ! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_283_power2__less__0,axiom,
    ! [A3: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_z3403309356797280102nteger ) ).

% power2_less_0
thf(fact_284_power2__less__0,axiom,
    ! [A3: real] :
      ~ ( ord_less_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).

% power2_less_0
thf(fact_285_power2__less__0,axiom,
    ! [A3: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_286_power2__less__0,axiom,
    ! [A3: int] :
      ~ ( ord_less_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).

% power2_less_0
thf(fact_287_power__strict__mono,axiom,
    ! [A3: code_integer,B3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_288_power__strict__mono,axiom,
    ! [A3: real,B3: real,N2: nat] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_289_power__strict__mono,axiom,
    ! [A3: rat,B3: rat,N2: nat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_290_power__strict__mono,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_291_power__strict__mono,axiom,
    ! [A3: int,B3: int,N2: nat] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_292_power2__less__imp__less,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
       => ( ord_le6747313008572928689nteger @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_293_power2__less__imp__less,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_294_power2__less__imp__less,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_295_power2__less__imp__less,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ord_less_nat @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_296_power2__less__imp__less,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_int @ X2 @ Y2 ) ) ) ).

% power2_less_imp_less
thf(fact_297_le__divide__eq__1__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
        = ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_298_le__divide__eq__1__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
        = ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% le_divide_eq_1_pos
thf(fact_299_le__divide__eq__1__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
        = ( ord_less_eq_real @ B3 @ A3 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_300_le__divide__eq__1__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
        = ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).

% le_divide_eq_1_neg
thf(fact_301_divide__le__eq__1__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
        = ( ord_less_eq_real @ B3 @ A3 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_302_divide__le__eq__1__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
        = ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).

% divide_le_eq_1_pos
thf(fact_303_divide__le__eq__1__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
        = ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_304_divide__le__eq__1__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
        = ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% divide_le_eq_1_neg
thf(fact_305_zero__less__divide__1__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A3 ) )
      = ( ord_less_real @ zero_zero_real @ A3 ) ) ).

% zero_less_divide_1_iff
thf(fact_306_zero__less__divide__1__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) )
      = ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).

% zero_less_divide_1_iff
thf(fact_307_less__divide__eq__1__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
        = ( ord_less_real @ A3 @ B3 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_308_less__divide__eq__1__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
        = ( ord_less_rat @ A3 @ B3 ) ) ) ).

% less_divide_eq_1_pos
thf(fact_309_less__divide__eq__1__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
        = ( ord_less_real @ B3 @ A3 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_310_less__divide__eq__1__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
        = ( ord_less_rat @ B3 @ A3 ) ) ) ).

% less_divide_eq_1_neg
thf(fact_311_divide__less__eq__1__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
        = ( ord_less_real @ B3 @ A3 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_312_divide__less__eq__1__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
        = ( ord_less_rat @ B3 @ A3 ) ) ) ).

% divide_less_eq_1_pos
thf(fact_313_divide__less__eq__1__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
        = ( ord_less_real @ A3 @ B3 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_314_divide__less__eq__1__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
        = ( ord_less_rat @ A3 @ B3 ) ) ) ).

% divide_less_eq_1_neg
thf(fact_315_divide__less__0__1__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A3 ) @ zero_zero_real )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_316_divide__less__0__1__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ zero_zero_rat )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_317_zero__le__divide__1__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% zero_le_divide_1_iff
thf(fact_318_zero__le__divide__1__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% zero_le_divide_1_iff
thf(fact_319_divide__le__0__1__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% divide_le_0_1_iff
thf(fact_320_divide__le__0__1__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A3 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_321_divide__eq__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ B3 )
        = zero_zero_real )
      = ( ( A3 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_322_divide__eq__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A3 @ B3 )
        = zero_zero_rat )
      = ( ( A3 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_323_divide__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( divide_divide_real @ C @ A3 )
        = ( divide_divide_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_324_divide__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ( divide_divide_rat @ C @ A3 )
        = ( divide_divide_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_left
thf(fact_325_divide__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ C )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_326_divide__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A3 @ C )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A3 = B3 ) ) ) ).

% divide_cancel_right
thf(fact_327_division__ring__divide__zero,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_328_division__ring__divide__zero,axiom,
    ! [A3: rat] :
      ( ( divide_divide_rat @ A3 @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_329_divide__eq__1__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ B3 )
        = one_one_real )
      = ( ( B3 != zero_zero_real )
        & ( A3 = B3 ) ) ) ).

% divide_eq_1_iff
thf(fact_330_divide__eq__1__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A3 @ B3 )
        = one_one_rat )
      = ( ( B3 != zero_zero_rat )
        & ( A3 = B3 ) ) ) ).

% divide_eq_1_iff
thf(fact_331_one__eq__divide__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A3 @ B3 ) )
      = ( ( B3 != zero_zero_real )
        & ( A3 = B3 ) ) ) ).

% one_eq_divide_iff
thf(fact_332_one__eq__divide__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A3 @ B3 ) )
      = ( ( B3 != zero_zero_rat )
        & ( A3 = B3 ) ) ) ).

% one_eq_divide_iff
thf(fact_333_divide__self,axiom,
    ! [A3: real] :
      ( ( A3 != zero_zero_real )
     => ( ( divide_divide_real @ A3 @ A3 )
        = one_one_real ) ) ).

% divide_self
thf(fact_334_divide__self,axiom,
    ! [A3: rat] :
      ( ( A3 != zero_zero_rat )
     => ( ( divide_divide_rat @ A3 @ A3 )
        = one_one_rat ) ) ).

% divide_self
thf(fact_335_divide__self__if,axiom,
    ! [A3: real] :
      ( ( ( A3 = zero_zero_real )
       => ( ( divide_divide_real @ A3 @ A3 )
          = zero_zero_real ) )
      & ( ( A3 != zero_zero_real )
       => ( ( divide_divide_real @ A3 @ A3 )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_336_divide__self__if,axiom,
    ! [A3: rat] :
      ( ( ( A3 = zero_zero_rat )
       => ( ( divide_divide_rat @ A3 @ A3 )
          = zero_zero_rat ) )
      & ( ( A3 != zero_zero_rat )
       => ( ( divide_divide_rat @ A3 @ A3 )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_337_divide__eq__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( ( divide_divide_real @ B3 @ A3 )
        = one_one_real )
      = ( ( A3 != zero_zero_real )
        & ( A3 = B3 ) ) ) ).

% divide_eq_eq_1
thf(fact_338_divide__eq__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ( divide_divide_rat @ B3 @ A3 )
        = one_one_rat )
      = ( ( A3 != zero_zero_rat )
        & ( A3 = B3 ) ) ) ).

% divide_eq_eq_1
thf(fact_339_eq__divide__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B3 @ A3 ) )
      = ( ( A3 != zero_zero_real )
        & ( A3 = B3 ) ) ) ).

% eq_divide_eq_1
thf(fact_340_eq__divide__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B3 @ A3 ) )
      = ( ( A3 != zero_zero_rat )
        & ( A3 = B3 ) ) ) ).

% eq_divide_eq_1
thf(fact_341_one__divide__eq__0__iff,axiom,
    ! [A3: real] :
      ( ( ( divide_divide_real @ one_one_real @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_342_one__divide__eq__0__iff,axiom,
    ! [A3: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A3 )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_343_zero__eq__1__divide__iff,axiom,
    ! [A3: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A3 ) )
      = ( A3 = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_344_zero__eq__1__divide__iff,axiom,
    ! [A3: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A3 ) )
      = ( A3 = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_345_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_346_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_347_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).

% linordered_field_no_lb
thf(fact_348_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X4 ) ).

% linordered_field_no_lb
thf(fact_349_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_350_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_351_divide__le__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).

% divide_le_0_iff
thf(fact_352_divide__le__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% divide_le_0_iff
thf(fact_353_divide__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_354_divide__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_right_mono
thf(fact_355_zero__le__divide__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_356_zero__le__divide__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_357_divide__nonneg__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_358_divide__nonneg__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_359_divide__nonneg__nonpos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_360_divide__nonneg__nonpos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_361_divide__nonpos__nonneg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_362_divide__nonpos__nonneg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_363_divide__nonpos__nonpos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_364_divide__nonpos__nonpos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_365_divide__right__mono__neg,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A3 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_366_divide__right__mono__neg,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( divide_divide_rat @ A3 @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_367_divide__neg__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_neg_neg
thf(fact_368_divide__neg__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_neg_neg
thf(fact_369_divide__neg__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_370_divide__neg__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_371_divide__pos__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_372_divide__pos__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_373_divide__pos__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_pos_pos
thf(fact_374_divide__pos__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_pos_pos
thf(fact_375_divide__less__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A3 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).

% divide_less_0_iff
thf(fact_376_divide__less__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A3 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% divide_less_0_iff
thf(fact_377_divide__less__cancel,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A3 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A3 ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_378_divide__less__cancel,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A3 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A3 ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_379_zero__less__divide__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_380_zero__less__divide__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_381_divide__strict__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_382_divide__strict__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_383_divide__strict__right__mono__neg,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_384_divide__strict__right__mono__neg,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_385_right__inverse__eq,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( ( divide_divide_real @ A3 @ B3 )
          = one_one_real )
        = ( A3 = B3 ) ) ) ).

% right_inverse_eq
thf(fact_386_right__inverse__eq,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A3 @ B3 )
          = one_one_rat )
        = ( A3 = B3 ) ) ) ).

% right_inverse_eq
thf(fact_387_frac__le,axiom,
    ! [Y2: real,X2: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_388_frac__le,axiom,
    ! [Y2: rat,X2: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_389_frac__less,axiom,
    ! [X2: real,Y2: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_390_frac__less,axiom,
    ! [X2: rat,Y2: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_391_frac__less2,axiom,
    ! [X2: real,Y2: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y2 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_392_frac__less2,axiom,
    ! [X2: rat,Y2: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ Y2 )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y2 @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_393_divide__le__cancel,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A3 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% divide_le_cancel
thf(fact_394_divide__le__cancel,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A3 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% divide_le_cancel
thf(fact_395_divide__nonneg__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_396_divide__nonneg__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_397_divide__nonneg__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_398_divide__nonneg__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_399_divide__nonpos__neg,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ Y2 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_400_divide__nonpos__neg,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_401_divide__nonpos__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_402_divide__nonpos__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_403_divide__less__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ B3 @ A3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ A3 @ B3 ) )
        | ( A3 = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_404_divide__less__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ B3 @ A3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ A3 @ B3 ) )
        | ( A3 = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_405_less__divide__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ A3 @ B3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ B3 @ A3 ) ) ) ) ).

% less_divide_eq_1
thf(fact_406_less__divide__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ A3 @ B3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A3 ) ) ) ) ).

% less_divide_eq_1
thf(fact_407_divide__le__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A3 ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ B3 @ A3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ A3 @ B3 ) )
        | ( A3 = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_408_divide__le__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A3 ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ B3 @ A3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ A3 @ B3 ) )
        | ( A3 = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_409_le__divide__eq__1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ A3 @ B3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% le_divide_eq_1
thf(fact_410_le__divide__eq__1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ A3 @ B3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% le_divide_eq_1
thf(fact_411_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_412_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_413_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_414_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_415_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_416_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_417_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_418_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_419_int__div__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ( divide_divide_int @ A3 @ B3 )
          = A3 )
        = ( B3 = one_one_int ) ) ) ).

% int_div_same_is_1
thf(fact_420_option_Ocollapse,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( ( some_P2407035485129114418it_nat @ ( the_Pr3501439614016493281it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_421_option_Ocollapse,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( ( some_P468703482102919278it_nat @ ( the_Pr5838048819577852031it_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_422_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_423_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_424_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_425_prod_Ocollapse,axiom,
    ! [Prod: product_prod_num_num] :
      ( ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_426_prod_Ocollapse,axiom,
    ! [Prod: produc8398139464844984134T_VEBT] :
      ( ( produc1750349459881913976T_VEBT @ ( produc758997459209783180T_VEBT @ Prod ) @ ( produc2084898568784432842T_VEBT @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_427_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_num] :
      ( ( product_Pair_nat_num @ ( product_fst_nat_num @ Prod ) @ ( product_snd_nat_num @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_428_prod_Ocollapse,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_429_prod_Ocollapse,axiom,
    ! [Prod: product_prod_int_int] :
      ( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_430_prod_Ocollapse,axiom,
    ! [Prod: produc6575502325842934193n_assn] :
      ( ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ Prod ) @ ( produc2051961928117032727n_assn @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_431_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_432_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_433_div__self,axiom,
    ! [A3: real] :
      ( ( A3 != zero_zero_real )
     => ( ( divide_divide_real @ A3 @ A3 )
        = one_one_real ) ) ).

% div_self
thf(fact_434_div__self,axiom,
    ! [A3: rat] :
      ( ( A3 != zero_zero_rat )
     => ( ( divide_divide_rat @ A3 @ A3 )
        = one_one_rat ) ) ).

% div_self
thf(fact_435_div__self,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( divide_divide_nat @ A3 @ A3 )
        = one_one_nat ) ) ).

% div_self
thf(fact_436_div__self,axiom,
    ! [A3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( divide_divide_int @ A3 @ A3 )
        = one_one_int ) ) ).

% div_self
thf(fact_437_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_438_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_439_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_440_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_441_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_442_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_443_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_444_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_445_two__pow__div__gt__le,axiom,
    ! [V: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ V @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% two_pow_div_gt_le
thf(fact_446_vebt__succi_Osimps,axiom,
    ( vEBT_vebt_succi
    = ( ^ [T: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ^ [Mima: product_prod_nat_nat] :
                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                    @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                        @ ^ [L: nat] :
                            ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                            @ ^ [H: nat] :
                                ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                @ ^ [Aktnode: vEBT_VEBTi] :
                                    ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                    @ ^ [Maxlow: option_nat] :
                                        ( if_Hea5867803462524415986on_nat
                                        @ ( ( Maxlow != none_nat )
                                          & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                        @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_succi @ Aktnode @ L )
                                          @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                        @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_succi @ Summary2 @ H )
                                          @ ^ [Succsum: option_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                    ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                    @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ Info2 )
          @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
          @ T ) ) ) ).

% vebt_succi.simps
thf(fact_447_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_448_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_449_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_rat @ M )
        = ( numeral_numeral_rat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_450_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_451_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_452_old_Oprod_Oinject,axiom,
    ! [A3: num,B3: num,A5: num,B4: num] :
      ( ( ( product_Pair_num_num @ A3 @ B3 )
        = ( product_Pair_num_num @ A5 @ B4 ) )
      = ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_453_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: produc4813437837504472865T_VEBT,A5: nat,B4: produc4813437837504472865T_VEBT] :
      ( ( ( produc1750349459881913976T_VEBT @ A3 @ B3 )
        = ( produc1750349459881913976T_VEBT @ A5 @ B4 ) )
      = ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_454_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: num,A5: nat,B4: num] :
      ( ( ( product_Pair_nat_num @ A3 @ B3 )
        = ( product_Pair_nat_num @ A5 @ B4 ) )
      = ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_455_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: nat,A5: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A5 @ B4 ) )
      = ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_456_old_Oprod_Oinject,axiom,
    ! [A3: int,B3: int,A5: int,B4: int] :
      ( ( ( product_Pair_int_int @ A3 @ B3 )
        = ( product_Pair_int_int @ A5 @ B4 ) )
      = ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_457_prod_Oinject,axiom,
    ! [X1: num,X22: num,Y1: num,Y22: num] :
      ( ( ( product_Pair_num_num @ X1 @ X22 )
        = ( product_Pair_num_num @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_458_prod_Oinject,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT,Y1: nat,Y22: produc4813437837504472865T_VEBT] :
      ( ( ( produc1750349459881913976T_VEBT @ X1 @ X22 )
        = ( produc1750349459881913976T_VEBT @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_459_prod_Oinject,axiom,
    ! [X1: nat,X22: num,Y1: nat,Y22: num] :
      ( ( ( product_Pair_nat_num @ X1 @ X22 )
        = ( product_Pair_nat_num @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_460_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_461_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_462_option_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_463_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_464_option_Oinject,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% option.inject
thf(fact_465_div__by__0,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_466_div__by__0,axiom,
    ! [A3: rat] :
      ( ( divide_divide_rat @ A3 @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_467_div__by__0,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ A3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_468_div__by__0,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ A3 @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_469_div__0,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ zero_zero_real @ A3 )
      = zero_zero_real ) ).

% div_0
thf(fact_470_div__0,axiom,
    ! [A3: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A3 )
      = zero_zero_rat ) ).

% div_0
thf(fact_471_div__0,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% div_0
thf(fact_472_div__0,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ zero_zero_int @ A3 )
      = zero_zero_int ) ).

% div_0
thf(fact_473_div__by__1,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ one_one_real )
      = A3 ) ).

% div_by_1
thf(fact_474_div__by__1,axiom,
    ! [A3: rat] :
      ( ( divide_divide_rat @ A3 @ one_one_rat )
      = A3 ) ).

% div_by_1
thf(fact_475_div__by__1,axiom,
    ! [A3: nat] :
      ( ( divide_divide_nat @ A3 @ one_one_nat )
      = A3 ) ).

% div_by_1
thf(fact_476_div__by__1,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ A3 @ one_one_int )
      = A3 ) ).

% div_by_1
thf(fact_477_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_478_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_479_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_480_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_481_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).

% bot_nat_0.extremum
thf(fact_482_not__Some__eq,axiom,
    ! [X2: option2621746655072343315it_nat] :
      ( ( ! [Y: produc120671012495760973it_nat] :
            ( X2
           != ( some_P2407035485129114418it_nat @ Y ) ) )
      = ( X2 = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq
thf(fact_483_not__Some__eq,axiom,
    ! [X2: option7339022715339332451it_nat] :
      ( ( ! [Y: produc8047831477865546771it_nat] :
            ( X2
           != ( some_P468703482102919278it_nat @ Y ) ) )
      = ( X2 = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq
thf(fact_484_not__Some__eq,axiom,
    ! [X2: option_nat] :
      ( ( ! [Y: nat] :
            ( X2
           != ( some_nat @ Y ) ) )
      = ( X2 = none_nat ) ) ).

% not_Some_eq
thf(fact_485_not__Some__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( ! [Y: product_prod_nat_nat] :
            ( X2
           != ( some_P7363390416028606310at_nat @ Y ) ) )
      = ( X2 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_486_not__Some__eq,axiom,
    ! [X2: option_num] :
      ( ( ! [Y: num] :
            ( X2
           != ( some_num @ Y ) ) )
      = ( X2 = none_num ) ) ).

% not_Some_eq
thf(fact_487_not__None__eq,axiom,
    ! [X2: option2621746655072343315it_nat] :
      ( ( X2 != none_P1551326421579882414it_nat )
      = ( ? [Y: produc120671012495760973it_nat] :
            ( X2
            = ( some_P2407035485129114418it_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_488_not__None__eq,axiom,
    ! [X2: option7339022715339332451it_nat] :
      ( ( X2 != none_P7668321371905463026it_nat )
      = ( ? [Y: produc8047831477865546771it_nat] :
            ( X2
            = ( some_P468703482102919278it_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_489_not__None__eq,axiom,
    ! [X2: option_nat] :
      ( ( X2 != none_nat )
      = ( ? [Y: nat] :
            ( X2
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_490_not__None__eq,axiom,
    ! [X2: option4927543243414619207at_nat] :
      ( ( X2 != none_P5556105721700978146at_nat )
      = ( ? [Y: product_prod_nat_nat] :
            ( X2
            = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_491_not__None__eq,axiom,
    ! [X2: option_num] :
      ( ( X2 != none_num )
      = ( ? [Y: num] :
            ( X2
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_492_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat,A3: nat,B3: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A3 @ B3 ) )
      = ( F @ A3 @ B3 ) ) ).

% case_prod_conv
thf(fact_493_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A3: nat,B3: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A3 @ B3 ) )
      = ( F @ A3 @ B3 ) ) ).

% case_prod_conv
thf(fact_494_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A3: int,B3: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A3 @ B3 ) )
      = ( F @ A3 @ B3 ) ) ).

% case_prod_conv
thf(fact_495_case__prod__conv,axiom,
    ! [F: int > int > $o,A3: int,B3: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A3 @ B3 ) )
      = ( F @ A3 @ B3 ) ) ).

% case_prod_conv
thf(fact_496_case__prod__conv,axiom,
    ! [F: int > int > int,A3: int,B3: int] :
      ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A3 @ B3 ) )
      = ( F @ A3 @ B3 ) ) ).

% case_prod_conv
thf(fact_497_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_498_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_499_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_500_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_501_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_502_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_503_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_504_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_505_vebt__succi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_succi @ Ti @ X2 ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) ) ).

% vebt_succi_refines
thf(fact_506_linorder__neqE__linordered__idom,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_507_linorder__neqE__linordered__idom,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_rat @ X2 @ Y2 )
       => ( ord_less_rat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_508_linorder__neqE__linordered__idom,axiom,
    ! [X2: int,Y2: int] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_int @ X2 @ Y2 )
       => ( ord_less_int @ Y2 @ X2 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_509_le__num__One__iff,axiom,
    ! [X2: num] :
      ( ( ord_less_eq_num @ X2 @ one )
      = ( X2 = one ) ) ).

% le_num_One_iff
thf(fact_510_Pair__inject,axiom,
    ! [A3: num,B3: num,A5: num,B4: num] :
      ( ( ( product_Pair_num_num @ A3 @ B3 )
        = ( product_Pair_num_num @ A5 @ B4 ) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_511_Pair__inject,axiom,
    ! [A3: nat,B3: produc4813437837504472865T_VEBT,A5: nat,B4: produc4813437837504472865T_VEBT] :
      ( ( ( produc1750349459881913976T_VEBT @ A3 @ B3 )
        = ( produc1750349459881913976T_VEBT @ A5 @ B4 ) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_512_Pair__inject,axiom,
    ! [A3: nat,B3: num,A5: nat,B4: num] :
      ( ( ( product_Pair_nat_num @ A3 @ B3 )
        = ( product_Pair_nat_num @ A5 @ B4 ) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_513_Pair__inject,axiom,
    ! [A3: nat,B3: nat,A5: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A5 @ B4 ) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_514_Pair__inject,axiom,
    ! [A3: int,B3: int,A5: int,B4: int] :
      ( ( ( product_Pair_int_int @ A3 @ B3 )
        = ( product_Pair_int_int @ A5 @ B4 ) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_515_prod__cases,axiom,
    ! [P: product_prod_num_num > $o,P2: product_prod_num_num] :
      ( ! [A: num,B: num] : ( P @ ( product_Pair_num_num @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_516_prod__cases,axiom,
    ! [P: produc8398139464844984134T_VEBT > $o,P2: produc8398139464844984134T_VEBT] :
      ( ! [A: nat,B: produc4813437837504472865T_VEBT] : ( P @ ( produc1750349459881913976T_VEBT @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_517_prod__cases,axiom,
    ! [P: product_prod_nat_num > $o,P2: product_prod_nat_num] :
      ( ! [A: nat,B: num] : ( P @ ( product_Pair_nat_num @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_518_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A: nat,B: nat] : ( P @ ( product_Pair_nat_nat @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_519_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
      ( ! [A: int,B: int] : ( P @ ( product_Pair_int_int @ A @ B ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_520_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X3: num,Y3: num] :
      ( P2
      = ( product_Pair_num_num @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_521_surj__pair,axiom,
    ! [P2: produc8398139464844984134T_VEBT] :
    ? [X3: nat,Y3: produc4813437837504472865T_VEBT] :
      ( P2
      = ( produc1750349459881913976T_VEBT @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_522_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X3: nat,Y3: num] :
      ( P2
      = ( product_Pair_nat_num @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_523_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_524_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P2
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_525_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_num_num] :
      ~ ! [A: num,B: num] :
          ( Y2
         != ( product_Pair_num_num @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_526_old_Oprod_Oexhaust,axiom,
    ! [Y2: produc8398139464844984134T_VEBT] :
      ~ ! [A: nat,B: produc4813437837504472865T_VEBT] :
          ( Y2
         != ( produc1750349459881913976T_VEBT @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_527_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_num] :
      ~ ! [A: nat,B: num] :
          ( Y2
         != ( product_Pair_nat_num @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_528_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_nat_nat] :
      ~ ! [A: nat,B: nat] :
          ( Y2
         != ( product_Pair_nat_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_529_old_Oprod_Oexhaust,axiom,
    ! [Y2: product_prod_int_int] :
      ~ ! [A: int,B: int] :
          ( Y2
         != ( product_Pair_int_int @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_530_linorder__neqE__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( X2 != Y2 )
     => ( ~ ( ord_less_nat @ X2 @ Y2 )
       => ( ord_less_nat @ Y2 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_531_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ~ ( P @ N4 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N4 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_532_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_533_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_534_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( ( ord_less_nat @ S @ T2 )
     => ( S != T2 ) ) ).

% less_not_refl3
thf(fact_535_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_536_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_537_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_538_size__neq__size__imp__neq,axiom,
    ! [X2: list_real,Y2: list_real] :
      ( ( ( size_size_list_real @ X2 )
       != ( size_size_list_real @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_539_size__neq__size__imp__neq,axiom,
    ! [X2: list_o,Y2: list_o] :
      ( ( ( size_size_list_o @ X2 )
       != ( size_size_list_o @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_540_size__neq__size__imp__neq,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X2 )
       != ( size_size_list_nat @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_541_size__neq__size__imp__neq,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( ( size_size_list_int @ X2 )
       != ( size_size_list_int @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_542_size__neq__size__imp__neq,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( size_size_num @ X2 )
       != ( size_size_num @ Y2 ) )
     => ( X2 != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_543_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ? [X3: nat] :
            ( ( P @ X3 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_544_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_545_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_546_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_547_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_548_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_549_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X23: nat] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_550_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_551_prod_Ocase__distrib,axiom,
    ! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_552_prod_Ocase__distrib,axiom,
    ! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_553_prod_Ocase__distrib,axiom,
    ! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_554_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc2626176000494625587at_nat @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X12: nat,X23: nat] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_555_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_nat_nat,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X12: nat,X23: nat] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_556_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_557_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc8211389475949308722nt_int
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_558_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X12: int,X23: int] : ( H2 @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_559_option_Ocase__distrib,axiom,
    ! [H2: num > num,F1: num,F2: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_560_option_Ocase__distrib,axiom,
    ! [H2: num > int,F1: num,F2: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_561_option_Ocase__distrib,axiom,
    ! [H2: int > num,F1: int,F2: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_562_option_Ocase__distrib,axiom,
    ! [H2: int > int,F1: int,F2: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_563_option_Ocase__distrib,axiom,
    ! [H2: option_num > num,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_option_num_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_564_option_Ocase__distrib,axiom,
    ! [H2: option_num > int,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_option_int_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_565_option_Ocase__distrib,axiom,
    ! [H2: num > option_num,F1: num,F2: num > num,Option: option_num] :
      ( ( H2 @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_566_option_Ocase__distrib,axiom,
    ! [H2: int > option_num,F1: int,F2: num > int,Option: option_num] :
      ( ( H2 @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_567_option_Ocase__distrib,axiom,
    ! [H2: option_num > option_num,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( H2 @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( case_o6005452278849405969um_num @ ( H2 @ F1 )
        @ ^ [X: num] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_568_option_Ocase__distrib,axiom,
    ! [H2: $o > $o,F1: $o,F2: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( H2 @ ( case_o184042715313410164at_nat @ F1 @ F2 @ Option ) )
      = ( case_o184042715313410164at_nat @ ( H2 @ F1 )
        @ ^ [X: product_prod_nat_nat] : ( H2 @ ( F2 @ X ) )
        @ Option ) ) ).

% option.case_distrib
thf(fact_569_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_570_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_571_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_572_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_573_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_574_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_575_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_576_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_577_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_578_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_579_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_580_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_581_zero__neq__one,axiom,
    zero_zero_uint32 != one_one_uint32 ).

% zero_neq_one
thf(fact_582_zero__neq__one,axiom,
    zero_z3563351764282998399l_num1 != one_on7727431528512463931l_num1 ).

% zero_neq_one
thf(fact_583_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_584_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_585_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_586_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_587_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_588_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_589_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_590_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_591_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_592_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_593_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_594_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_595_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P @ N4 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N4 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_596_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_597_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_598_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_599_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_600_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_601_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_602_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_603_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_604_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_605_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_606_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_607_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_608_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_609_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N: nat] :
          ( ( ord_less_nat @ M3 @ N )
          | ( M3 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_610_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_611_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N: nat] :
          ( ( ord_less_eq_nat @ M3 @ N )
          & ( M3 != N ) ) ) ) ).

% nat_less_le
thf(fact_612_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat,X1: nat,X22: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_613_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X22: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_614_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_615_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_616_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_617_fst__conv,axiom,
    ! [X1: num,X22: num] :
      ( ( product_fst_num_num @ ( product_Pair_num_num @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_618_fst__conv,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT] :
      ( ( produc758997459209783180T_VEBT @ ( produc1750349459881913976T_VEBT @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_619_fst__conv,axiom,
    ! [X1: nat,X22: num] :
      ( ( product_fst_nat_num @ ( product_Pair_nat_num @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_620_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_621_fst__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_622_fst__conv,axiom,
    ! [X1: assn,X22: assn] :
      ( ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_623_fst__eqD,axiom,
    ! [X2: num,Y2: num,A3: num] :
      ( ( ( product_fst_num_num @ ( product_Pair_num_num @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_624_fst__eqD,axiom,
    ! [X2: nat,Y2: produc4813437837504472865T_VEBT,A3: nat] :
      ( ( ( produc758997459209783180T_VEBT @ ( produc1750349459881913976T_VEBT @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_625_fst__eqD,axiom,
    ! [X2: nat,Y2: num,A3: nat] :
      ( ( ( product_fst_nat_num @ ( product_Pair_nat_num @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_626_fst__eqD,axiom,
    ! [X2: nat,Y2: nat,A3: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_627_fst__eqD,axiom,
    ! [X2: int,Y2: int,A3: int] :
      ( ( ( product_fst_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_628_fst__eqD,axiom,
    ! [X2: assn,Y2: assn,A3: assn] :
      ( ( ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) )
        = A3 )
     => ( X2 = A3 ) ) ).

% fst_eqD
thf(fact_629_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: nat,B: nat] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y2
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_630_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: nat,B: num] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y2
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_631_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: num,B: nat] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y2
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_632_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: num,B: num] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y2
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_633_combine__options__cases,axiom,
    ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: nat,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_nat @ A ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_634_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: product_prod_nat_nat,B: nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y2
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_635_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y2: option_num] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_num )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: product_prod_nat_nat,B: num] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y2
                  = ( some_num @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_636_combine__options__cases,axiom,
    ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_num )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: num,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_num @ A ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_637_combine__options__cases,axiom,
    ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y2: option4927543243414619207at_nat] :
      ( ( ( X2 = none_P5556105721700978146at_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_P5556105721700978146at_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ A ) )
             => ( ( Y2
                  = ( some_P7363390416028606310at_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_638_combine__options__cases,axiom,
    ! [X2: option7339022715339332451it_nat,P: option7339022715339332451it_nat > option_nat > $o,Y2: option_nat] :
      ( ( ( X2 = none_P7668321371905463026it_nat )
       => ( P @ X2 @ Y2 ) )
     => ( ( ( Y2 = none_nat )
         => ( P @ X2 @ Y2 ) )
       => ( ! [A: produc8047831477865546771it_nat,B: nat] :
              ( ( X2
                = ( some_P468703482102919278it_nat @ A ) )
             => ( ( Y2
                  = ( some_nat @ B ) )
               => ( P @ X2 @ Y2 ) ) )
         => ( P @ X2 @ Y2 ) ) ) ) ).

% combine_options_cases
thf(fact_639_split__option__all,axiom,
    ( ( ^ [P3: option2621746655072343315it_nat > $o] :
        ! [X5: option2621746655072343315it_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option2621746655072343315it_nat > $o] :
          ( ( P4 @ none_P1551326421579882414it_nat )
          & ! [X: produc120671012495760973it_nat] : ( P4 @ ( some_P2407035485129114418it_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_640_split__option__all,axiom,
    ( ( ^ [P3: option7339022715339332451it_nat > $o] :
        ! [X5: option7339022715339332451it_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option7339022715339332451it_nat > $o] :
          ( ( P4 @ none_P7668321371905463026it_nat )
          & ! [X: produc8047831477865546771it_nat] : ( P4 @ ( some_P468703482102919278it_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_641_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X5: option_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X: nat] : ( P4 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_642_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X5: option4927543243414619207at_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_643_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X5: option_num] : ( P3 @ X5 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X: num] : ( P4 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_644_split__option__ex,axiom,
    ( ( ^ [P3: option2621746655072343315it_nat > $o] :
        ? [X5: option2621746655072343315it_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option2621746655072343315it_nat > $o] :
          ( ( P4 @ none_P1551326421579882414it_nat )
          | ? [X: produc120671012495760973it_nat] : ( P4 @ ( some_P2407035485129114418it_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_645_split__option__ex,axiom,
    ( ( ^ [P3: option7339022715339332451it_nat > $o] :
        ? [X5: option7339022715339332451it_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option7339022715339332451it_nat > $o] :
          ( ( P4 @ none_P7668321371905463026it_nat )
          | ? [X: produc8047831477865546771it_nat] : ( P4 @ ( some_P468703482102919278it_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_646_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X5: option_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X: nat] : ( P4 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_647_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X5: option4927543243414619207at_nat] : ( P3 @ X5 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_648_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X5: option_num] : ( P3 @ X5 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X: num] : ( P4 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_649_option_Oexhaust,axiom,
    ! [Y2: option2621746655072343315it_nat] :
      ( ( Y2 != none_P1551326421579882414it_nat )
     => ~ ! [X24: produc120671012495760973it_nat] :
            ( Y2
           != ( some_P2407035485129114418it_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_650_option_Oexhaust,axiom,
    ! [Y2: option7339022715339332451it_nat] :
      ( ( Y2 != none_P7668321371905463026it_nat )
     => ~ ! [X24: produc8047831477865546771it_nat] :
            ( Y2
           != ( some_P468703482102919278it_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_651_option_Oexhaust,axiom,
    ! [Y2: option_nat] :
      ( ( Y2 != none_nat )
     => ~ ! [X24: nat] :
            ( Y2
           != ( some_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_652_option_Oexhaust,axiom,
    ! [Y2: option4927543243414619207at_nat] :
      ( ( Y2 != none_P5556105721700978146at_nat )
     => ~ ! [X24: product_prod_nat_nat] :
            ( Y2
           != ( some_P7363390416028606310at_nat @ X24 ) ) ) ).

% option.exhaust
thf(fact_653_option_Oexhaust,axiom,
    ! [Y2: option_num] :
      ( ( Y2 != none_num )
     => ~ ! [X24: num] :
            ( Y2
           != ( some_num @ X24 ) ) ) ).

% option.exhaust
thf(fact_654_option_OdiscI,axiom,
    ! [Option: option2621746655072343315it_nat,X22: produc120671012495760973it_nat] :
      ( ( Option
        = ( some_P2407035485129114418it_nat @ X22 ) )
     => ( Option != none_P1551326421579882414it_nat ) ) ).

% option.discI
thf(fact_655_option_OdiscI,axiom,
    ! [Option: option7339022715339332451it_nat,X22: produc8047831477865546771it_nat] :
      ( ( Option
        = ( some_P468703482102919278it_nat @ X22 ) )
     => ( Option != none_P7668321371905463026it_nat ) ) ).

% option.discI
thf(fact_656_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_657_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_658_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_659_option_Odistinct_I1_J,axiom,
    ! [X22: produc120671012495760973it_nat] :
      ( none_P1551326421579882414it_nat
     != ( some_P2407035485129114418it_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_660_option_Odistinct_I1_J,axiom,
    ! [X22: produc8047831477865546771it_nat] :
      ( none_P7668321371905463026it_nat
     != ( some_P468703482102919278it_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_661_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_662_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_663_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_664_snd__conv,axiom,
    ! [X1: num,X22: num] :
      ( ( product_snd_num_num @ ( product_Pair_num_num @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_665_snd__conv,axiom,
    ! [X1: nat,X22: produc4813437837504472865T_VEBT] :
      ( ( produc2084898568784432842T_VEBT @ ( produc1750349459881913976T_VEBT @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_666_snd__conv,axiom,
    ! [X1: nat,X22: num] :
      ( ( product_snd_nat_num @ ( product_Pair_nat_num @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_667_snd__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_668_snd__conv,axiom,
    ! [X1: int,X22: int] :
      ( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_669_snd__conv,axiom,
    ! [X1: assn,X22: assn] :
      ( ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_670_snd__eqD,axiom,
    ! [X2: num,Y2: num,A3: num] :
      ( ( ( product_snd_num_num @ ( product_Pair_num_num @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_671_snd__eqD,axiom,
    ! [X2: nat,Y2: produc4813437837504472865T_VEBT,A3: produc4813437837504472865T_VEBT] :
      ( ( ( produc2084898568784432842T_VEBT @ ( produc1750349459881913976T_VEBT @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_672_snd__eqD,axiom,
    ! [X2: nat,Y2: num,A3: num] :
      ( ( ( product_snd_nat_num @ ( product_Pair_nat_num @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_673_snd__eqD,axiom,
    ! [X2: nat,Y2: nat,A3: nat] :
      ( ( ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_674_snd__eqD,axiom,
    ! [X2: int,Y2: int,A3: int] :
      ( ( ( product_snd_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_675_snd__eqD,axiom,
    ! [X2: assn,Y2: assn,A3: assn] :
      ( ( ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) )
        = A3 )
     => ( Y2 = A3 ) ) ).

% snd_eqD
thf(fact_676_prod__induct3,axiom,
    ! [P: produc8398139464844984134T_VEBT > $o,X2: produc8398139464844984134T_VEBT] :
      ( ! [A: nat,B: list_VEBT_VEBT,C2: vEBT_VEBT] : ( P @ ( produc1750349459881913976T_VEBT @ A @ ( produc6691630295680060561T_VEBT @ B @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_677_prod__cases3,axiom,
    ! [Y2: produc8398139464844984134T_VEBT] :
      ~ ! [A: nat,B: list_VEBT_VEBT,C2: vEBT_VEBT] :
          ( Y2
         != ( produc1750349459881913976T_VEBT @ A @ ( produc6691630295680060561T_VEBT @ B @ C2 ) ) ) ).

% prod_cases3
thf(fact_678_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_679_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_680_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_681_prod__eq__iff,axiom,
    ( ( ^ [Y5: product_prod_nat_nat,Z2: product_prod_nat_nat] : Y5 = Z2 )
    = ( ^ [S2: product_prod_nat_nat,T: product_prod_nat_nat] :
          ( ( ( product_fst_nat_nat @ S2 )
            = ( product_fst_nat_nat @ T ) )
          & ( ( product_snd_nat_nat @ S2 )
            = ( product_snd_nat_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_682_prod__eq__iff,axiom,
    ( ( ^ [Y5: product_prod_int_int,Z2: product_prod_int_int] : Y5 = Z2 )
    = ( ^ [S2: product_prod_int_int,T: product_prod_int_int] :
          ( ( ( product_fst_int_int @ S2 )
            = ( product_fst_int_int @ T ) )
          & ( ( product_snd_int_int @ S2 )
            = ( product_snd_int_int @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_683_prod__eq__iff,axiom,
    ( ( ^ [Y5: produc6575502325842934193n_assn,Z2: produc6575502325842934193n_assn] : Y5 = Z2 )
    = ( ^ [S2: produc6575502325842934193n_assn,T: produc6575502325842934193n_assn] :
          ( ( ( produc9167289414957590229n_assn @ S2 )
            = ( produc9167289414957590229n_assn @ T ) )
          & ( ( produc2051961928117032727n_assn @ S2 )
            = ( produc2051961928117032727n_assn @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_684_prod__eqI,axiom,
    ! [P2: product_prod_nat_nat,Q2: product_prod_nat_nat] :
      ( ( ( product_fst_nat_nat @ P2 )
        = ( product_fst_nat_nat @ Q2 ) )
     => ( ( ( product_snd_nat_nat @ P2 )
          = ( product_snd_nat_nat @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_685_prod__eqI,axiom,
    ! [P2: product_prod_int_int,Q2: product_prod_int_int] :
      ( ( ( product_fst_int_int @ P2 )
        = ( product_fst_int_int @ Q2 ) )
     => ( ( ( product_snd_int_int @ P2 )
          = ( product_snd_int_int @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_686_prod__eqI,axiom,
    ! [P2: produc6575502325842934193n_assn,Q2: produc6575502325842934193n_assn] :
      ( ( ( produc9167289414957590229n_assn @ P2 )
        = ( produc9167289414957590229n_assn @ Q2 ) )
     => ( ( ( produc2051961928117032727n_assn @ P2 )
          = ( produc2051961928117032727n_assn @ Q2 ) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
thf(fact_687_prod_Oexpand,axiom,
    ! [Prod: product_prod_nat_nat,Prod2: product_prod_nat_nat] :
      ( ( ( ( product_fst_nat_nat @ Prod )
          = ( product_fst_nat_nat @ Prod2 ) )
        & ( ( product_snd_nat_nat @ Prod )
          = ( product_snd_nat_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_688_prod_Oexpand,axiom,
    ! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
      ( ( ( ( product_fst_int_int @ Prod )
          = ( product_fst_int_int @ Prod2 ) )
        & ( ( product_snd_int_int @ Prod )
          = ( product_snd_int_int @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_689_prod_Oexpand,axiom,
    ! [Prod: produc6575502325842934193n_assn,Prod2: produc6575502325842934193n_assn] :
      ( ( ( ( produc9167289414957590229n_assn @ Prod )
          = ( produc9167289414957590229n_assn @ Prod2 ) )
        & ( ( produc2051961928117032727n_assn @ Prod )
          = ( produc2051961928117032727n_assn @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_690_option_Oexpand,axiom,
    ! [Option: option2621746655072343315it_nat,Option2: option2621746655072343315it_nat] :
      ( ( ( Option = none_P1551326421579882414it_nat )
        = ( Option2 = none_P1551326421579882414it_nat ) )
     => ( ( ( Option != none_P1551326421579882414it_nat )
         => ( ( Option2 != none_P1551326421579882414it_nat )
           => ( ( the_Pr3501439614016493281it_nat @ Option )
              = ( the_Pr3501439614016493281it_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_691_option_Oexpand,axiom,
    ! [Option: option7339022715339332451it_nat,Option2: option7339022715339332451it_nat] :
      ( ( ( Option = none_P7668321371905463026it_nat )
        = ( Option2 = none_P7668321371905463026it_nat ) )
     => ( ( ( Option != none_P7668321371905463026it_nat )
         => ( ( Option2 != none_P7668321371905463026it_nat )
           => ( ( the_Pr5838048819577852031it_nat @ Option )
              = ( the_Pr5838048819577852031it_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_692_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_693_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_694_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_695_option_Osimps_I5_J,axiom,
    ! [F1: heap_Time_Heap_o,F2: product_prod_nat_nat > heap_Time_Heap_o,X22: product_prod_nat_nat] :
      ( ( case_o1442776274061689234at_nat @ F1 @ F2 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( F2 @ X22 ) ) ).

% option.simps(5)
thf(fact_696_option_Osimps_I5_J,axiom,
    ! [F1: option_num,F2: num > option_num,X22: num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F2 @ ( some_num @ X22 ) )
      = ( F2 @ X22 ) ) ).

% option.simps(5)
thf(fact_697_option_Osimps_I5_J,axiom,
    ! [F1: $o,F2: product_prod_nat_nat > $o,X22: product_prod_nat_nat] :
      ( ( case_o184042715313410164at_nat @ F1 @ F2 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( F2 @ X22 ) ) ).

% option.simps(5)
thf(fact_698_option_Osimps_I5_J,axiom,
    ! [F1: num,F2: num > num,X22: num] :
      ( ( case_option_num_num @ F1 @ F2 @ ( some_num @ X22 ) )
      = ( F2 @ X22 ) ) ).

% option.simps(5)
thf(fact_699_option_Osimps_I5_J,axiom,
    ! [F1: int,F2: num > int,X22: num] :
      ( ( case_option_int_num @ F1 @ F2 @ ( some_num @ X22 ) )
      = ( F2 @ X22 ) ) ).

% option.simps(5)
thf(fact_700_option_Osimps_I4_J,axiom,
    ! [F1: heap_Time_Heap_o,F2: product_prod_nat_nat > heap_Time_Heap_o] :
      ( ( case_o1442776274061689234at_nat @ F1 @ F2 @ none_P5556105721700978146at_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_701_option_Osimps_I4_J,axiom,
    ! [F1: option_num,F2: num > option_num] :
      ( ( case_o6005452278849405969um_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_702_option_Osimps_I4_J,axiom,
    ! [F1: $o,F2: product_prod_nat_nat > $o] :
      ( ( case_o184042715313410164at_nat @ F1 @ F2 @ none_P5556105721700978146at_nat )
      = F1 ) ).

% option.simps(4)
thf(fact_703_option_Osimps_I4_J,axiom,
    ! [F1: num,F2: num > num] :
      ( ( case_option_num_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_704_option_Osimps_I4_J,axiom,
    ! [F1: int,F2: num > int] :
      ( ( case_option_int_num @ F1 @ F2 @ none_num )
      = F1 ) ).

% option.simps(4)
thf(fact_705_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc2626176000494625587at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_706_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_707_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_708_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_709_cond__case__prod__eta,axiom,
    ! [F: int > int > int,G: product_prod_int_int > int] :
      ( ! [X3: int,Y3: int] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_int_int @ X3 @ Y3 ) ) )
     => ( ( produc8211389475949308722nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_710_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_711_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_712_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_713_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_714_case__prod__eta,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_715_case__prodE2,axiom,
    ! [Q: product_prod_nat_nat > $o,P: nat > nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc2626176000494625587at_nat @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_716_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_717_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_718_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_719_case__prodE2,axiom,
    ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
      ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( Z
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( Q @ ( P @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_720_fst__def,axiom,
    ( product_fst_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X12: nat,X23: nat] : X12 ) ) ).

% fst_def
thf(fact_721_fst__def,axiom,
    ( produc9167289414957590229n_assn
    = ( produc2152611005075324454n_assn
      @ ^ [X12: assn,X23: assn] : X12 ) ) ).

% fst_def
thf(fact_722_fst__def,axiom,
    ( product_fst_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X12: int,X23: int] : X12 ) ) ).

% fst_def
thf(fact_723_snd__def,axiom,
    ( product_snd_nat_nat
    = ( produc6842872674320459806at_nat
      @ ^ [X12: nat,X23: nat] : X23 ) ) ).

% snd_def
thf(fact_724_snd__def,axiom,
    ( produc2051961928117032727n_assn
    = ( produc2152611005075324454n_assn
      @ ^ [X12: assn,X23: assn] : X23 ) ) ).

% snd_def
thf(fact_725_snd__def,axiom,
    ( product_snd_int_int
    = ( produc8211389475949308722nt_int
      @ ^ [X12: int,X23: int] : X23 ) ) ).

% snd_def
thf(fact_726_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_727_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_728_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_729_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_730_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_731_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_732_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_733_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_734_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_735_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_736_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_737_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_738_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_739_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_740_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_741_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_742_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_743_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_744_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_745_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_755_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_756_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_757_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_758_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_759_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_760_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_761_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_762_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_763_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_764_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_765_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_766_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_767_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_768_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_769_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_770_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_771_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_772_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_773_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_774_numeral__One,axiom,
    ( ( numera9087168376688890119uint32 @ one )
    = one_one_uint32 ) ).

% numeral_One
thf(fact_775_numeral__One,axiom,
    ( ( numera7442385471795722001l_num1 @ one )
    = one_on7727431528512463931l_num1 ) ).

% numeral_One
thf(fact_776_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_777_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_778_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_779_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_780_divide__numeral__1,axiom,
    ! [A3: real] :
      ( ( divide_divide_real @ A3 @ ( numeral_numeral_real @ one ) )
      = A3 ) ).

% divide_numeral_1
thf(fact_781_divide__numeral__1,axiom,
    ! [A3: rat] :
      ( ( divide_divide_rat @ A3 @ ( numeral_numeral_rat @ one ) )
      = A3 ) ).

% divide_numeral_1
thf(fact_782_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_783_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_784_surjective__pairing,axiom,
    ! [T2: product_prod_num_num] :
      ( T2
      = ( product_Pair_num_num @ ( product_fst_num_num @ T2 ) @ ( product_snd_num_num @ T2 ) ) ) ).

% surjective_pairing
thf(fact_785_surjective__pairing,axiom,
    ! [T2: produc8398139464844984134T_VEBT] :
      ( T2
      = ( produc1750349459881913976T_VEBT @ ( produc758997459209783180T_VEBT @ T2 ) @ ( produc2084898568784432842T_VEBT @ T2 ) ) ) ).

% surjective_pairing
thf(fact_786_surjective__pairing,axiom,
    ! [T2: product_prod_nat_num] :
      ( T2
      = ( product_Pair_nat_num @ ( product_fst_nat_num @ T2 ) @ ( product_snd_nat_num @ T2 ) ) ) ).

% surjective_pairing
thf(fact_787_surjective__pairing,axiom,
    ! [T2: product_prod_nat_nat] :
      ( T2
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ T2 ) @ ( product_snd_nat_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_788_surjective__pairing,axiom,
    ! [T2: product_prod_int_int] :
      ( T2
      = ( product_Pair_int_int @ ( product_fst_int_int @ T2 ) @ ( product_snd_int_int @ T2 ) ) ) ).

% surjective_pairing
thf(fact_789_surjective__pairing,axiom,
    ! [T2: produc6575502325842934193n_assn] :
      ( T2
      = ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ T2 ) @ ( produc2051961928117032727n_assn @ T2 ) ) ) ).

% surjective_pairing
thf(fact_790_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_num_num] :
      ( Prod
      = ( product_Pair_num_num @ ( product_fst_num_num @ Prod ) @ ( product_snd_num_num @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_791_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8398139464844984134T_VEBT] :
      ( Prod
      = ( produc1750349459881913976T_VEBT @ ( produc758997459209783180T_VEBT @ Prod ) @ ( produc2084898568784432842T_VEBT @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_792_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_num] :
      ( Prod
      = ( product_Pair_nat_num @ ( product_fst_nat_num @ Prod ) @ ( product_snd_nat_num @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_793_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( Prod
      = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_794_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_int_int] :
      ( Prod
      = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_795_prod_Oexhaust__sel,axiom,
    ! [Prod: produc6575502325842934193n_assn] :
      ( Prod
      = ( produc118845697133431529n_assn @ ( produc9167289414957590229n_assn @ Prod ) @ ( produc2051961928117032727n_assn @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_796_option_Oexhaust__sel,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
     => ( Option
        = ( some_P2407035485129114418it_nat @ ( the_Pr3501439614016493281it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_797_option_Oexhaust__sel,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
     => ( Option
        = ( some_P468703482102919278it_nat @ ( the_Pr5838048819577852031it_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_798_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_799_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_800_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_801_case__prod__beta,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [F4: nat > nat > product_prod_nat_nat,P5: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_802_case__prod__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F4: nat > nat > $o,P5: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_803_case__prod__beta,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F4: int > int > product_prod_int_int,P5: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_804_case__prod__beta,axiom,
    ( produc4947309494688390418_int_o
    = ( ^ [F4: int > int > $o,P5: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_805_case__prod__beta,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F4: int > int > int,P5: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_beta
thf(fact_806_split__beta,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [F4: nat > nat > product_prod_nat_nat,Prod3: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_807_split__beta,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F4: nat > nat > $o,Prod3: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ Prod3 ) @ ( product_snd_nat_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_808_split__beta,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F4: int > int > product_prod_int_int,Prod3: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_809_split__beta,axiom,
    ( produc4947309494688390418_int_o
    = ( ^ [F4: int > int > $o,Prod3: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_810_split__beta,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F4: int > int > int,Prod3: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ Prod3 ) @ ( product_snd_int_int @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_811_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
        = ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_812_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A3 ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_813_int__div__less__self,axiom,
    ! [X2: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).

% int_div_less_self
thf(fact_814_div__neg__pos__less0,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_815_option_Ocase__eq__if,axiom,
    ( case_o1442776274061689234at_nat
    = ( ^ [F13: heap_Time_Heap_o,F23: product_prod_nat_nat > heap_Time_Heap_o,Option3: option4927543243414619207at_nat] : ( if_Heap_Time_Heap_o @ ( Option3 = none_P5556105721700978146at_nat ) @ F13 @ ( F23 @ ( the_Pr8591224930841456533at_nat @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_816_option_Ocase__eq__if,axiom,
    ( case_o6005452278849405969um_num
    = ( ^ [F13: option_num,F23: num > option_num,Option3: option_num] : ( if_option_num @ ( Option3 = none_num ) @ F13 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_817_option_Ocase__eq__if,axiom,
    ( case_o184042715313410164at_nat
    = ( ^ [F13: $o,F23: product_prod_nat_nat > $o,Option3: option4927543243414619207at_nat] :
          ( ( ( Option3 = none_P5556105721700978146at_nat )
           => F13 )
          & ( ( Option3 != none_P5556105721700978146at_nat )
           => ( F23 @ ( the_Pr8591224930841456533at_nat @ Option3 ) ) ) ) ) ) ).

% option.case_eq_if
thf(fact_818_option_Ocase__eq__if,axiom,
    ( case_option_num_num
    = ( ^ [F13: num,F23: num > num,Option3: option_num] : ( if_num @ ( Option3 = none_num ) @ F13 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_819_option_Ocase__eq__if,axiom,
    ( case_option_int_num
    = ( ^ [F13: int,F23: num > int,Option3: option_num] : ( if_int @ ( Option3 = none_num ) @ F13 @ ( F23 @ ( the_num @ Option3 ) ) ) ) ) ).

% option.case_eq_if
thf(fact_820_case__prod__unfold,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [C3: nat > nat > product_prod_nat_nat,P5: product_prod_nat_nat] : ( C3 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_821_case__prod__unfold,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [C3: nat > nat > $o,P5: product_prod_nat_nat] : ( C3 @ ( product_fst_nat_nat @ P5 ) @ ( product_snd_nat_nat @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_822_case__prod__unfold,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [C3: int > int > product_prod_int_int,P5: product_prod_int_int] : ( C3 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_823_case__prod__unfold,axiom,
    ( produc4947309494688390418_int_o
    = ( ^ [C3: int > int > $o,P5: product_prod_int_int] : ( C3 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_824_case__prod__unfold,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [C3: int > int > int,P5: product_prod_int_int] : ( C3 @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ).

% case_prod_unfold
thf(fact_825_case__prod__beta_H,axiom,
    ( produc2626176000494625587at_nat
    = ( ^ [F4: nat > nat > product_prod_nat_nat,X: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_826_case__prod__beta_H,axiom,
    ( produc6081775807080527818_nat_o
    = ( ^ [F4: nat > nat > $o,X: product_prod_nat_nat] : ( F4 @ ( product_fst_nat_nat @ X ) @ ( product_snd_nat_nat @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_827_case__prod__beta_H,axiom,
    ( produc4245557441103728435nt_int
    = ( ^ [F4: int > int > product_prod_int_int,X: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_828_case__prod__beta_H,axiom,
    ( produc4947309494688390418_int_o
    = ( ^ [F4: int > int > $o,X: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_829_case__prod__beta_H,axiom,
    ( produc8211389475949308722nt_int
    = ( ^ [F4: int > int > int,X: product_prod_int_int] : ( F4 @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).

% case_prod_beta'
thf(fact_830_option_Othe__def,axiom,
    ( the_ui685118366354182287uint32
    = ( case_o6709414378691970003uint32 @ undefi332904144742839227uint32
      @ ^ [X23: ( uint32 > uint32 > uint32 ) > uint32 > uint32 > uint32] : X23 ) ) ).

% option.the_def
thf(fact_831_option_Othe__def,axiom,
    ( the_ui5136145761085816068eger_o
    = ( case_o4437601675458612413eger_o @ undefi6981832269580975664eger_o
      @ ^ [X23: ( uint32 > nat > $o ) > uint32 > code_integer > $o] : X23 ) ) ).

% option.the_def
thf(fact_832_option_Othe__def,axiom,
    ( the_ui8720505876773817540uint32
    = ( case_o6228893485755354685uint32 @ undefi8537048349889504752uint32
      @ ^ [X23: ( uint32 > nat > $o > uint32 ) > uint32 > code_integer > $o > uint32] : X23 ) ) ).

% option.the_def
thf(fact_833_option_Othe__def,axiom,
    ( the_na2292640131888687716uint32
    = ( case_o8336680350232271869uint32 @ undefi7330133036835070352uint32
      @ ^ [X23: ( nat > uint32 > uint32 ) > code_integer > uint32 > uint32] : X23 ) ) ).

% option.the_def
thf(fact_834_option_Othe__def,axiom,
    ( the_na3915024202274359524uint32
    = ( case_o6516889040143735037uint32 @ undefi8952517107220742160uint32
      @ ^ [X23: ( nat > uint32 > uint32 ) > uint32 > code_integer > uint32] : X23 ) ) ).

% option.the_def
thf(fact_835_option_Othe__def,axiom,
    ( the_nat
    = ( case_option_nat_nat @ undefined_nat
      @ ^ [X23: nat] : X23 ) ) ).

% option.the_def
thf(fact_836_option_Othe__def,axiom,
    ( the_Pr8591224930841456533at_nat
    = ( case_o7430979018509204427at_nat @ undefi3946296454836805481at_nat
      @ ^ [X23: product_prod_nat_nat] : X23 ) ) ).

% option.the_def
thf(fact_837_option_Othe__def,axiom,
    ( the_num
    = ( case_option_num_num @ undefined_num
      @ ^ [X23: num] : X23 ) ) ).

% option.the_def
thf(fact_838_div__positive,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A3 )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_positive
thf(fact_839_div__positive,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ B3 @ A3 )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_positive
thf(fact_840_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ A3 @ B3 )
       => ( ( divide_divide_nat @ A3 @ B3 )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_841_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ A3 @ B3 )
       => ( ( divide_divide_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_842_prod_Osplit__sel__asm,axiom,
    ! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_843_prod_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_844_prod_Osplit__sel__asm,axiom,
    ! [P: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( P @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_845_prod_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( P @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_846_prod_Osplit__sel__asm,axiom,
    ! [P: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( P @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
            & ~ ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_847_prod_Osplit__sel,axiom,
    ! [P: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc2626176000494625587at_nat @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_848_prod_Osplit__sel,axiom,
    ! [P: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( P @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_nat_nat @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_nat_nat @ Prod ) @ ( product_snd_nat_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_849_prod_Osplit__sel,axiom,
    ! [P: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( P @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_850_prod_Osplit__sel,axiom,
    ! [P: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( P @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_851_prod_Osplit__sel,axiom,
    ! [P: int > $o,F: int > int > int,Prod: product_prod_int_int] :
      ( ( P @ ( produc8211389475949308722nt_int @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) )
       => ( P @ ( F @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_852_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
        = ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_853_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_854_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) )
        = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_855_zdiv__le__dividend,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ A3 ) ) ) ).

% zdiv_le_dividend
thf(fact_856_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_857_div__nonpos__pos__le0,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_858_div__nonneg__neg__le0,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_859_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_860_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_861_zdiv__mono2__neg,axiom,
    ! [A3: int,B4: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B3 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B4 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_862_zdiv__mono1__neg,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ A5 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A5 @ B3 ) @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_863_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_864_zdiv__mono2,axiom,
    ! [A3: int,B4: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B3 )
         => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A3 @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_865_zdiv__mono1,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ A5 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ A5 @ B3 ) ) ) ) ).

% zdiv_mono1
thf(fact_866_option_Osplit__sel__asm,axiom,
    ! [P: heap_Time_Heap_o > $o,F1: heap_Time_Heap_o,F2: product_prod_nat_nat > heap_Time_Heap_o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o1442776274061689234at_nat @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_P5556105721700978146at_nat )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
              & ~ ( P @ ( F2 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_867_option_Osplit__sel__asm,axiom,
    ! [P: option_num > $o,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_868_option_Osplit__sel__asm,axiom,
    ! [P: $o > $o,F1: $o,F2: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o184042715313410164at_nat @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_P5556105721700978146at_nat )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
              & ~ ( P @ ( F2 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_869_option_Osplit__sel__asm,axiom,
    ! [P: num > $o,F1: num,F2: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_870_option_Osplit__sel__asm,axiom,
    ! [P: int > $o,F1: int,F2: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( ~ ( ( ( Option = none_num )
              & ~ ( P @ F1 ) )
            | ( ( Option
                = ( some_num @ ( the_num @ Option ) ) )
              & ~ ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ) ).

% option.split_sel_asm
thf(fact_871_option_Osplit__sel,axiom,
    ! [P: heap_Time_Heap_o > $o,F1: heap_Time_Heap_o,F2: product_prod_nat_nat > heap_Time_Heap_o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o1442776274061689234at_nat @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_P5556105721700978146at_nat )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
         => ( P @ ( F2 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_872_option_Osplit__sel,axiom,
    ! [P: option_num > $o,F1: option_num,F2: num > option_num,Option: option_num] :
      ( ( P @ ( case_o6005452278849405969um_num @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_873_option_Osplit__sel,axiom,
    ! [P: $o > $o,F1: $o,F2: product_prod_nat_nat > $o,Option: option4927543243414619207at_nat] :
      ( ( P @ ( case_o184042715313410164at_nat @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_P5556105721700978146at_nat )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) )
         => ( P @ ( F2 @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_874_option_Osplit__sel,axiom,
    ! [P: num > $o,F1: num,F2: num > num,Option: option_num] :
      ( ( P @ ( case_option_num_num @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_875_option_Osplit__sel,axiom,
    ! [P: int > $o,F1: int,F2: num > int,Option: option_num] :
      ( ( P @ ( case_option_int_num @ F1 @ F2 @ Option ) )
      = ( ( ( Option = none_num )
         => ( P @ F1 ) )
        & ( ( Option
            = ( some_num @ ( the_num @ Option ) ) )
         => ( P @ ( F2 @ ( the_num @ Option ) ) ) ) ) ) ).

% option.split_sel
thf(fact_876_half__gt__zero__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( ord_less_real @ zero_zero_real @ A3 ) ) ).

% half_gt_zero_iff
thf(fact_877_half__gt__zero__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).

% half_gt_zero_iff
thf(fact_878_half__gt__zero,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_879_half__gt__zero,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_880_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_881_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_882_pred__list__to__short,axiom,
    ! [Deg4: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_883_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ( vEBT_VEBT_vebt_succi
    = ( ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6250501799366334488on_nat
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
              @ ^ [Uu: product_unit] :
                  ( produc2190226783763740553on_nat
                  @ ^ [Info3: option4927543243414619207at_nat] :
                      ( produc2606485630176857543on_nat
                      @ ^ [Deg3: nat] :
                          ( produc6124225815318652422on_nat
                          @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                              ( heap_T5999496708990702694on_nat
                              @ ( refine_Imp_assert
                                @ ( ( Info3 = Info2 )
                                  & ( Deg3 = Deg2 )
                                  & ( vEBT_is_Node @ T ) ) )
                              @ ^ [Uv: product_unit] :
                                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                  @ ^ [Mima: product_prod_nat_nat] :
                                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                                        @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L: nat] :
                                                ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_T5999496708990702694on_nat
                                                    @ ( refine_Imp_assert
                                                      @ ( L
                                                        = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    @ ^ [Uw: product_unit] :
                                                        ( heap_T5999496708990702694on_nat
                                                        @ ( refine_Imp_assert
                                                          @ ( H
                                                            = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                        @ ^ [Ux: product_unit] :
                                                            ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                            @ ^ [Uy: product_unit] :
                                                                ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                @ ^ [Aktnode: vEBT_VEBTi] :
                                                                    ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                                                    @ ^ [Maxlow: option_nat] :
                                                                        ( heap_T5999496708990702694on_nat
                                                                        @ ( refine_Imp_assert
                                                                          @ ( Maxlow
                                                                            = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                        @ ^ [Uz: product_unit] :
                                                                            ( if_Hea5867803462524415986on_nat
                                                                            @ ( ( Maxlow != none_nat )
                                                                              & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                                            @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_succi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                              @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                            @ ( heap_T3669509953089699273on_nat @ ( vEBT_VEBT_vebt_succi @ Summary3 @ Summary2 @ H )
                                                                              @ ^ [Succsum: option_nat] :
                                                                                  ( heap_T5999496708990702694on_nat
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( Succsum = none_nat )
                                                                                      = ( ( vEBT_vebt_succ @ Summary3 @ H )
                                                                                        = none_nat ) ) )
                                                                                  @ ^ [Va: product_unit] :
                                                                                      ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                                                            @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ Info2 ) ) ) )
                  @ ( vEBT_c634343235235684882T_VEBT
                    @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                    @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                    @ T ) ) )
          @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_succi'.simps
thf(fact_884_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_885_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_886_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_887_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(71)
thf(fact_888_pred__max,axiom,
    ! [Deg4: nat,Ma: nat,X2: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_889_less__option__None__Some__code,axiom,
    ! [X2: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X2 ) ) ).

% less_option_None_Some_code
thf(fact_890_less__option__None__Some__code,axiom,
    ! [X2: num] : ( ord_less_option_num @ none_num @ ( some_num @ X2 ) ) ).

% less_option_None_Some_code
thf(fact_891_less__eq__option__Some__None,axiom,
    ! [X2: nat] :
      ~ ( ord_le5914376470875661696on_nat @ ( some_nat @ X2 ) @ none_nat ) ).

% less_eq_option_Some_None
thf(fact_892_less__eq__option__Some__None,axiom,
    ! [X2: num] :
      ~ ( ord_le6622620407824499402on_num @ ( some_num @ X2 ) @ none_num ) ).

% less_eq_option_Some_None
thf(fact_893_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_894_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_895_case__prodI,axiom,
    ! [F: num > num > $o,A3: num,B3: num] :
      ( ( F @ A3 @ B3 )
     => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ).

% case_prodI
thf(fact_896_case__prodI,axiom,
    ! [F: nat > produc4813437837504472865T_VEBT > $o,A3: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( F @ A3 @ B3 )
     => ( produc2834603712688810931VEBT_o @ F @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) ) ) ).

% case_prodI
thf(fact_897_case__prodI,axiom,
    ! [F: nat > num > $o,A3: nat,B3: num] :
      ( ( F @ A3 @ B3 )
     => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ).

% case_prodI
thf(fact_898_case__prodI,axiom,
    ! [F: nat > nat > $o,A3: nat,B3: nat] :
      ( ( F @ A3 @ B3 )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ).

% case_prodI
thf(fact_899_case__prodI,axiom,
    ! [F: int > int > $o,A3: int,B3: int] :
      ( ( F @ A3 @ B3 )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).

% case_prodI
thf(fact_900_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: num > num > $o] :
      ( ! [A: num,B: num] :
          ( ( P2
            = ( product_Pair_num_num @ A @ B ) )
         => ( C @ A @ B ) )
     => ( produc5703948589228662326_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_901_case__prodI2,axiom,
    ! [P2: produc8398139464844984134T_VEBT,C: nat > produc4813437837504472865T_VEBT > $o] :
      ( ! [A: nat,B: produc4813437837504472865T_VEBT] :
          ( ( P2
            = ( produc1750349459881913976T_VEBT @ A @ B ) )
         => ( C @ A @ B ) )
     => ( produc2834603712688810931VEBT_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_902_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: nat > num > $o] :
      ( ! [A: nat,B: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( C @ A @ B ) )
     => ( produc4927758841916487424_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_903_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A @ B ) )
         => ( C @ A @ B ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_904_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: int > int > $o] :
      ( ! [A: int,B: int] :
          ( ( P2
            = ( product_Pair_int_int @ A @ B ) )
         => ( C @ A @ B ) )
     => ( produc4947309494688390418_int_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_905_mem__case__prodI,axiom,
    ! [Z: nat,C: num > num > set_nat,A3: num,B3: num] :
      ( ( member_nat @ Z @ ( C @ A3 @ B3 ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_906_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: num > num > set_VEBT_VEBT,A3: num,B3: num] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B3 ) )
     => ( member_VEBT_VEBT @ Z @ ( produc1023323404773863986T_VEBT @ C @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_907_mem__case__prodI,axiom,
    ! [Z: int,C: num > num > set_int,A3: num,B3: num] :
      ( ( member_int @ Z @ ( C @ A3 @ B3 ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_908_mem__case__prodI,axiom,
    ! [Z: real,C: num > num > set_real,A3: num,B3: num] :
      ( ( member_real @ Z @ ( C @ A3 @ B3 ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_909_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > num > set_nat,A3: nat,B3: num] :
      ( ( member_nat @ Z @ ( C @ A3 @ B3 ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_910_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: nat > num > set_VEBT_VEBT,A3: nat,B3: num] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B3 ) )
     => ( member_VEBT_VEBT @ Z @ ( produc7751347746043978236T_VEBT @ C @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_911_mem__case__prodI,axiom,
    ! [Z: int,C: nat > num > set_int,A3: nat,B3: num] :
      ( ( member_int @ Z @ ( C @ A3 @ B3 ) )
     => ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_912_mem__case__prodI,axiom,
    ! [Z: real,C: nat > num > set_real,A3: nat,B3: num] :
      ( ( member_real @ Z @ ( C @ A3 @ B3 ) )
     => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_913_mem__case__prodI,axiom,
    ! [Z: nat,C: nat > nat > set_nat,A3: nat,B3: nat] :
      ( ( member_nat @ Z @ ( C @ A3 @ B3 ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_914_mem__case__prodI,axiom,
    ! [Z: vEBT_VEBT,C: nat > nat > set_VEBT_VEBT,A3: nat,B3: nat] :
      ( ( member_VEBT_VEBT @ Z @ ( C @ A3 @ B3 ) )
     => ( member_VEBT_VEBT @ Z @ ( produc7709906084162714054T_VEBT @ C @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ) ).

% mem_case_prodI
thf(fact_915_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: nat,C: num > num > set_nat] :
      ( ! [A: num,B: num] :
          ( ( P2
            = ( product_Pair_num_num @ A @ B ) )
         => ( member_nat @ Z @ ( C @ A @ B ) ) )
     => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_916_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: vEBT_VEBT,C: num > num > set_VEBT_VEBT] :
      ( ! [A: num,B: num] :
          ( ( P2
            = ( product_Pair_num_num @ A @ B ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc1023323404773863986T_VEBT @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_917_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: int,C: num > num > set_int] :
      ( ! [A: num,B: num] :
          ( ( P2
            = ( product_Pair_num_num @ A @ B ) )
         => ( member_int @ Z @ ( C @ A @ B ) ) )
     => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_918_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z: real,C: num > num > set_real] :
      ( ! [A: num,B: num] :
          ( ( P2
            = ( product_Pair_num_num @ A @ B ) )
         => ( member_real @ Z @ ( C @ A @ B ) ) )
     => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_919_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: nat,C: nat > num > set_nat] :
      ( ! [A: nat,B: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( member_nat @ Z @ ( C @ A @ B ) ) )
     => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_920_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: vEBT_VEBT,C: nat > num > set_VEBT_VEBT] :
      ( ! [A: nat,B: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc7751347746043978236T_VEBT @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_921_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: int,C: nat > num > set_int] :
      ( ! [A: nat,B: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( member_int @ Z @ ( C @ A @ B ) ) )
     => ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_922_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z: real,C: nat > num > set_real] :
      ( ! [A: nat,B: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( member_real @ Z @ ( C @ A @ B ) ) )
     => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_923_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: nat,C: nat > nat > set_nat] :
      ( ! [A: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A @ B ) )
         => ( member_nat @ Z @ ( C @ A @ B ) ) )
     => ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_924_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z: vEBT_VEBT,C: nat > nat > set_VEBT_VEBT] :
      ( ! [A: nat,B: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A @ B ) )
         => ( member_VEBT_VEBT @ Z @ ( C @ A @ B ) ) )
     => ( member_VEBT_VEBT @ Z @ ( produc7709906084162714054T_VEBT @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_925_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_926_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_927_less__eq__option__None__code,axiom,
    ! [X2: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X2 ) ).

% less_eq_option_None_code
thf(fact_928_less__eq__option__None__code,axiom,
    ! [X2: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X2 ) ).

% less_eq_option_None_code
thf(fact_929_succ__min,axiom,
    ! [Deg4: nat,X2: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_930_less__option__None,axiom,
    ! [X2: option_num] :
      ~ ( ord_less_option_num @ X2 @ none_num ) ).

% less_option_None
thf(fact_931_less__option__None,axiom,
    ! [X2: option_nat] :
      ~ ( ord_less_option_nat @ X2 @ none_nat ) ).

% less_option_None
thf(fact_932_succ__list__to__short,axiom,
    ! [Deg4: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_933_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_934_less__eq__option__Some,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ X2 ) @ ( some_set_nat @ Y2 ) )
      = ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_935_less__eq__option__Some,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ X2 ) @ ( some_rat @ Y2 ) )
      = ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_936_less__eq__option__Some,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ X2 ) @ ( some_num @ Y2 ) )
      = ( ord_less_eq_num @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_937_less__eq__option__Some,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
      = ( ord_less_eq_nat @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_938_less__eq__option__Some,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ X2 ) @ ( some_int @ Y2 ) )
      = ( ord_less_eq_int @ X2 @ Y2 ) ) ).

% less_eq_option_Some
thf(fact_939_less__option__Some,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_option_real @ ( some_real @ X2 ) @ ( some_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% less_option_Some
thf(fact_940_less__option__Some,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_option_rat @ ( some_rat @ X2 ) @ ( some_rat @ Y2 ) )
      = ( ord_less_rat @ X2 @ Y2 ) ) ).

% less_option_Some
thf(fact_941_less__option__Some,axiom,
    ! [X2: num,Y2: num] :
      ( ( ord_less_option_num @ ( some_num @ X2 ) @ ( some_num @ Y2 ) )
      = ( ord_less_num @ X2 @ Y2 ) ) ).

% less_option_Some
thf(fact_942_less__option__Some,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_option_nat @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
      = ( ord_less_nat @ X2 @ Y2 ) ) ).

% less_option_Some
thf(fact_943_less__option__Some,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_option_int @ ( some_int @ X2 ) @ ( some_int @ Y2 ) )
      = ( ord_less_int @ X2 @ Y2 ) ) ).

% less_option_Some
thf(fact_944_succ__greatereq__min,axiom,
    ! [Deg4: nat,Mi: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_945_succ__less__length__list,axiom,
    ! [Deg4: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ Mi @ X2 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( 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 @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_946_pred__less__length__list,axiom,
    ! [Deg4: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_947_pred__lesseq__max,axiom,
    ! [Deg4: nat,X2: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ( ord_less_eq_nat @ X2 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_948_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_949_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_950_less__eq__option__def,axiom,
    ( ord_le2843612097646854710et_nat
    = ( ^ [X: option_set_nat,Y: option_set_nat] :
          ( case_o4401850862724306899et_nat @ $true
          @ ^ [Z3: set_nat] : ( case_o4401850862724306899et_nat @ $false @ ( ord_less_eq_set_nat @ Z3 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_951_less__eq__option__def,axiom,
    ( ord_le2406147912482264968on_rat
    = ( ^ [X: option_rat,Y: option_rat] :
          ( case_option_o_rat @ $true
          @ ^ [Z3: rat] : ( case_option_o_rat @ $false @ ( ord_less_eq_rat @ Z3 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_952_less__eq__option__def,axiom,
    ( ord_le6622620407824499402on_num
    = ( ^ [X: option_num,Y: option_num] :
          ( case_option_o_num @ $true
          @ ^ [Z3: num] : ( case_option_o_num @ $false @ ( ord_less_eq_num @ Z3 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_953_less__eq__option__def,axiom,
    ( ord_le5914376470875661696on_nat
    = ( ^ [X: option_nat,Y: option_nat] :
          ( case_option_o_nat @ $true
          @ ^ [Z3: nat] : ( case_option_o_nat @ $false @ ( ord_less_eq_nat @ Z3 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_954_less__eq__option__def,axiom,
    ( ord_le1736525451366464988on_int
    = ( ^ [X: option_int,Y: option_int] :
          ( case_option_o_int @ $true
          @ ^ [Z3: int] : ( case_option_o_int @ $false @ ( ord_less_eq_int @ Z3 ) @ Y )
          @ X ) ) ) ).

% less_eq_option_def
thf(fact_955_less__option__def,axiom,
    ( ord_less_option_real
    = ( ^ [X: option_real] :
          ( case_option_o_real @ $false
          @ ^ [Y: real] :
              ( case_option_o_real @ $true
              @ ^ [Z3: real] : ( ord_less_real @ Z3 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_956_less__option__def,axiom,
    ( ord_less_option_rat
    = ( ^ [X: option_rat] :
          ( case_option_o_rat @ $false
          @ ^ [Y: rat] :
              ( case_option_o_rat @ $true
              @ ^ [Z3: rat] : ( ord_less_rat @ Z3 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_957_less__option__def,axiom,
    ( ord_less_option_num
    = ( ^ [X: option_num] :
          ( case_option_o_num @ $false
          @ ^ [Y: num] :
              ( case_option_o_num @ $true
              @ ^ [Z3: num] : ( ord_less_num @ Z3 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_958_less__option__def,axiom,
    ( ord_less_option_nat
    = ( ^ [X: option_nat] :
          ( case_option_o_nat @ $false
          @ ^ [Y: nat] :
              ( case_option_o_nat @ $true
              @ ^ [Z3: nat] : ( ord_less_nat @ Z3 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_959_less__option__def,axiom,
    ( ord_less_option_int
    = ( ^ [X: option_int] :
          ( case_option_o_int @ $false
          @ ^ [Y: int] :
              ( case_option_o_int @ $true
              @ ^ [Z3: int] : ( ord_less_int @ Z3 @ Y )
              @ X ) ) ) ) ).

% less_option_def
thf(fact_960_mem__case__prodE,axiom,
    ! [Z: nat,C: num > num > set_nat,P2: product_prod_num_num] :
      ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_961_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: num > num > set_VEBT_VEBT,P2: product_prod_num_num] :
      ( ( member_VEBT_VEBT @ Z @ ( produc1023323404773863986T_VEBT @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_962_mem__case__prodE,axiom,
    ! [Z: int,C: num > num > set_int,P2: product_prod_num_num] :
      ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_963_mem__case__prodE,axiom,
    ! [Z: real,C: num > num > set_real,P2: product_prod_num_num] :
      ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P2 ) )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_964_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > num > set_nat,P2: product_prod_nat_num] :
      ( ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_965_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: nat > num > set_VEBT_VEBT,P2: product_prod_nat_num] :
      ( ( member_VEBT_VEBT @ Z @ ( produc7751347746043978236T_VEBT @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_966_mem__case__prodE,axiom,
    ! [Z: int,C: nat > num > set_int,P2: product_prod_nat_num] :
      ( ( member_int @ Z @ ( produc9175805072616146554et_int @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_int @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_967_mem__case__prodE,axiom,
    ! [Z: real,C: nat > num > set_real,P2: product_prod_nat_num] :
      ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( member_real @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_968_mem__case__prodE,axiom,
    ! [Z: nat,C: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z @ ( produc6189476227299908564et_nat @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_969_mem__case__prodE,axiom,
    ! [Z: vEBT_VEBT,C: nat > nat > set_VEBT_VEBT,P2: product_prod_nat_nat] :
      ( ( member_VEBT_VEBT @ Z @ ( produc7709906084162714054T_VEBT @ C @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_VEBT_VEBT @ Z @ ( C @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_970_case__prodD,axiom,
    ! [F: num > num > $o,A3: num,B3: num] :
      ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A3 @ B3 ) )
     => ( F @ A3 @ B3 ) ) ).

% case_prodD
thf(fact_971_case__prodD,axiom,
    ! [F: nat > produc4813437837504472865T_VEBT > $o,A3: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( produc2834603712688810931VEBT_o @ F @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) )
     => ( F @ A3 @ B3 ) ) ).

% case_prodD
thf(fact_972_case__prodD,axiom,
    ! [F: nat > num > $o,A3: nat,B3: num] :
      ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( F @ A3 @ B3 ) ) ).

% case_prodD
thf(fact_973_case__prodD,axiom,
    ! [F: nat > nat > $o,A3: nat,B3: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( F @ A3 @ B3 ) ) ).

% case_prodD
thf(fact_974_case__prodD,axiom,
    ! [F: int > int > $o,A3: int,B3: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( F @ A3 @ B3 ) ) ).

% case_prodD
thf(fact_975_case__prodE,axiom,
    ! [C: num > num > $o,P2: product_prod_num_num] :
      ( ( produc5703948589228662326_num_o @ C @ P2 )
     => ~ ! [X3: num,Y3: num] :
            ( ( P2
              = ( product_Pair_num_num @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_976_case__prodE,axiom,
    ! [C: nat > produc4813437837504472865T_VEBT > $o,P2: produc8398139464844984134T_VEBT] :
      ( ( produc2834603712688810931VEBT_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: produc4813437837504472865T_VEBT] :
            ( ( P2
              = ( produc1750349459881913976T_VEBT @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_977_case__prodE,axiom,
    ! [C: nat > num > $o,P2: product_prod_nat_num] :
      ( ( produc4927758841916487424_num_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_978_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_979_case__prodE,axiom,
    ! [C: int > int > $o,P2: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P2 )
     => ~ ! [X3: int,Y3: int] :
            ( ( P2
              = ( product_Pair_int_int @ X3 @ Y3 ) )
           => ~ ( C @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_980_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option != none_P1551326421579882414it_nat )
      = ( case_o535201446637900608it_nat @ $false
        @ ^ [Uu: produc120671012495760973it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_981_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option != none_P7668321371905463026it_nat )
      = ( case_o1358941076187788256it_nat @ $false
        @ ^ [Uu: produc8047831477865546771it_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_982_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
      = ( case_option_o_num @ $false
        @ ^ [Uu: num] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_983_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
      = ( case_option_o_nat @ $false
        @ ^ [Uu: nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_984_option_Odisc__eq__case_I2_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
      = ( case_o184042715313410164at_nat @ $false
        @ ^ [Uu: product_prod_nat_nat] : $true
        @ Option ) ) ).

% option.disc_eq_case(2)
thf(fact_985_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option2621746655072343315it_nat] :
      ( ( Option = none_P1551326421579882414it_nat )
      = ( case_o535201446637900608it_nat @ $true
        @ ^ [Uu: produc120671012495760973it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_986_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option7339022715339332451it_nat] :
      ( ( Option = none_P7668321371905463026it_nat )
      = ( case_o1358941076187788256it_nat @ $true
        @ ^ [Uu: produc8047831477865546771it_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_987_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_num] :
      ( ( Option = none_num )
      = ( case_option_o_num @ $true
        @ ^ [Uu: num] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_988_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option_nat] :
      ( ( Option = none_nat )
      = ( case_option_o_nat @ $true
        @ ^ [Uu: nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_989_option_Odisc__eq__case_I1_J,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option = none_P5556105721700978146at_nat )
      = ( case_o184042715313410164at_nat @ $true
        @ ^ [Uu: product_prod_nat_nat] : $false
        @ Option ) ) ).

% option.disc_eq_case(1)
thf(fact_990_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: produc6575502325842934193n_assn,A4: assn > assn > $o] :
      ( ( member7957490590177025114n_assn @ X2 @ ( collec1604292450182575004n_assn @ ( produc7274209992780475162assn_o @ A4 ) ) )
     => ( A4 @ ( produc9167289414957590229n_assn @ X2 ) @ ( produc2051961928117032727n_assn @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_991_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: product_prod_nat_nat,A4: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ X2 @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) )
     => ( A4 @ ( product_fst_nat_nat @ X2 ) @ ( product_snd_nat_nat @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_992_Product__Type_OCollect__case__prodD,axiom,
    ! [X2: product_prod_int_int,A4: int > int > $o] :
      ( ( member5262025264175285858nt_int @ X2 @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) )
     => ( A4 @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_993_case__optionE,axiom,
    ! [P: $o,Q: produc120671012495760973it_nat > $o,X2: option2621746655072343315it_nat] :
      ( ( case_o535201446637900608it_nat @ P @ Q @ X2 )
     => ( ( ( X2 = none_P1551326421579882414it_nat )
         => ~ P )
       => ~ ! [Y3: produc120671012495760973it_nat] :
              ( ( X2
                = ( some_P2407035485129114418it_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_994_case__optionE,axiom,
    ! [P: $o,Q: produc8047831477865546771it_nat > $o,X2: option7339022715339332451it_nat] :
      ( ( case_o1358941076187788256it_nat @ P @ Q @ X2 )
     => ( ( ( X2 = none_P7668321371905463026it_nat )
         => ~ P )
       => ~ ! [Y3: produc8047831477865546771it_nat] :
              ( ( X2
                = ( some_P468703482102919278it_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_995_case__optionE,axiom,
    ! [P: $o,Q: nat > $o,X2: option_nat] :
      ( ( case_option_o_nat @ P @ Q @ X2 )
     => ( ( ( X2 = none_nat )
         => ~ P )
       => ~ ! [Y3: nat] :
              ( ( X2
                = ( some_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_996_case__optionE,axiom,
    ! [P: $o,Q: num > $o,X2: option_num] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( ( X2 = none_num )
         => ~ P )
       => ~ ! [Y3: num] :
              ( ( X2
                = ( some_num @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_997_case__optionE,axiom,
    ! [P: $o,Q: product_prod_nat_nat > $o,X2: option4927543243414619207at_nat] :
      ( ( case_o184042715313410164at_nat @ P @ Q @ X2 )
     => ( ( ( X2 = none_P5556105721700978146at_nat )
         => ~ P )
       => ~ ! [Y3: product_prod_nat_nat] :
              ( ( X2
                = ( some_P7363390416028606310at_nat @ Y3 ) )
             => ~ ( Q @ Y3 ) ) ) ) ).

% case_optionE
thf(fact_998_less__eq__option__None,axiom,
    ! [X2: option_num] : ( ord_le6622620407824499402on_num @ none_num @ X2 ) ).

% less_eq_option_None
thf(fact_999_less__eq__option__None,axiom,
    ! [X2: option_nat] : ( ord_le5914376470875661696on_nat @ none_nat @ X2 ) ).

% less_eq_option_None
thf(fact_1000_less__eq__option__None__is__None,axiom,
    ! [X2: option_num] :
      ( ( ord_le6622620407824499402on_num @ X2 @ none_num )
     => ( X2 = none_num ) ) ).

% less_eq_option_None_is_None
thf(fact_1001_less__eq__option__None__is__None,axiom,
    ! [X2: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ X2 @ none_nat )
     => ( X2 = none_nat ) ) ).

% less_eq_option_None_is_None
thf(fact_1002_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1003_less__option__None__Some,axiom,
    ! [X2: nat] : ( ord_less_option_nat @ none_nat @ ( some_nat @ X2 ) ) ).

% less_option_None_Some
thf(fact_1004_less__option__None__Some,axiom,
    ! [X2: num] : ( ord_less_option_num @ none_num @ ( some_num @ X2 ) ) ).

% less_option_None_Some
thf(fact_1005_less__option__None__is__Some,axiom,
    ! [X2: option_nat] :
      ( ( ord_less_option_nat @ none_nat @ X2 )
     => ? [Z4: nat] :
          ( X2
          = ( some_nat @ Z4 ) ) ) ).

% less_option_None_is_Some
thf(fact_1006_less__option__None__is__Some,axiom,
    ! [X2: option_num] :
      ( ( ord_less_option_num @ none_num @ X2 )
     => ? [Z4: num] :
          ( X2
          = ( some_num @ Z4 ) ) ) ).

% less_option_None_is_Some
thf(fact_1007_tdeletemimi_H,axiom,
    ! [Deg4: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 ) @ one_one_nat ) ) ).

% tdeletemimi'
thf(fact_1008_VEBT__internal_Ovebt__predi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o,Xa: produc3960855945107176009Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ! [Vebt_predi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: option_nat,N5: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_predi @ A6 @ B5 @ Ba ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ Ba @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                      @ ^ [Uu: product_unit] :
                          ( produc2190226783763740553on_nat
                          @ ^ [Info3: option4927543243414619207at_nat] :
                              ( produc2606485630176857543on_nat
                              @ ^ [Deg3: nat] :
                                  ( produc6124225815318652422on_nat
                                  @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                      ( heap_T5999496708990702694on_nat
                                      @ ( refine_Imp_assert
                                        @ ( ( Info3 = Info2 )
                                          & ( Deg3 = Deg2 )
                                          & ( vEBT_is_Node @ T3 ) ) )
                                      @ ^ [Uv: product_unit] :
                                          ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ^ [Mima: product_prod_nat_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [L: nat] :
                                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_T5999496708990702694on_nat
                                                          @ ( refine_Imp_assert
                                                            @ ( L
                                                              = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          @ ^ [Uw: product_unit] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Ux: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                  @ ^ [Uy: product_unit] :
                                                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                                                          ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                                                          @ ^ [Minlow: option_nat] :
                                                                              ( heap_T5999496708990702694on_nat
                                                                              @ ( refine_Imp_assert
                                                                                @ ( Minlow
                                                                                  = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                              @ ^ [Uz: product_unit] :
                                                                                  ( if_Hea5867803462524415986on_nat
                                                                                  @ ( ( Minlow != none_nat )
                                                                                    & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                    @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi @ Summary3 @ Summary2 @ H )
                                                                                    @ ^ [Predsum: option_nat] :
                                                                                        ( heap_T5999496708990702694on_nat
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Predsum = none_nat )
                                                                                            = ( ( vEBT_vebt_pred @ Summary3 @ H )
                                                                                              = none_nat ) ) )
                                                                                        @ ^ [Va: product_unit] :
                                                                                            ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                  ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                  @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info2 ) ) ) )
                          @ ( vEBT_c634343235235684882T_VEBT
                            @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                            @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                            @ T3 ) ) )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X3 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa2
                @ N4 )
             => ( P @ T3 @ Ti4 @ X3 @ Ta @ Tia @ Xa2 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc183673358652719894on_nat @ ( produc1061038227461121684on_nat @ vEBT_VEBT_vebt_predi ) @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc8313044543888072982_nat_o @ ( produc275000359906850836_nat_o @ P ) @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
thf(fact_1009_VEBT__internal_Ovebt__succi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o,Xa: produc3960855945107176009Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ! [Vebt_succi: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: option_nat,N5: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_succi @ A6 @ B5 @ Ba ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ Ba @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                      @ ^ [Uu: product_unit] :
                          ( produc2190226783763740553on_nat
                          @ ^ [Info3: option4927543243414619207at_nat] :
                              ( produc2606485630176857543on_nat
                              @ ^ [Deg3: nat] :
                                  ( produc6124225815318652422on_nat
                                  @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                      ( heap_T5999496708990702694on_nat
                                      @ ( refine_Imp_assert
                                        @ ( ( Info3 = Info2 )
                                          & ( Deg3 = Deg2 )
                                          & ( vEBT_is_Node @ T3 ) ) )
                                      @ ^ [Uv: product_unit] :
                                          ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                          @ ^ [Mima: product_prod_nat_nat] :
                                              ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X3 @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                  @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [L: nat] :
                                                        ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( L
                                                                = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Uw: product_unit] :
                                                                ( heap_T5999496708990702694on_nat
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Ux: product_unit] :
                                                                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                    @ ^ [Uy: product_unit] :
                                                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                                                            @ ^ [Maxlow: option_nat] :
                                                                                ( heap_T5999496708990702694on_nat
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( Maxlow
                                                                                    = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                @ ^ [Uz: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat
                                                                                    @ ( ( Maxlow != none_nat )
                                                                                      & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                      @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi @ Summary3 @ Summary2 @ H )
                                                                                      @ ^ [Succsum: option_nat] :
                                                                                          ( heap_T5999496708990702694on_nat
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Succsum = none_nat )
                                                                                              = ( ( vEBT_vebt_succ @ Summary3 @ H )
                                                                                                = none_nat ) ) )
                                                                                          @ ^ [Va: product_unit] :
                                                                                              ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                    ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                    @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                          @ Info2 ) ) ) )
                          @ ( vEBT_c634343235235684882T_VEBT
                            @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                            @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                            @ T3 ) ) )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X3 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa2
                @ N4 )
             => ( P @ T3 @ Ti4 @ X3 @ Ta @ Tia @ Xa2 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc183673358652719894on_nat @ ( produc1061038227461121684on_nat @ vEBT_VEBT_vebt_succi ) @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc8313044543888072982_nat_o @ ( produc275000359906850836_nat_o @ P ) @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
thf(fact_1010_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
        & ( ( X2 = Mi )
          | ( X2 = Ma )
          | ( ( ord_less_nat @ X2 @ Ma )
            & ( ord_less_nat @ Mi @ X2 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_1011_insert__simp__mima,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        | ( X2 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ).

% insert_simp_mima
thf(fact_1012_vebt__predi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o,Xa: produc3881548065746020326Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ! [Vebt_predi2: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: option_nat,N5: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_predi2 @ A6 @ B5 ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ^ [Mima: product_prod_nat_nat] :
                          ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                            @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [L: nat] :
                                  ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                          ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                          @ ^ [Minlow: option_nat] :
                                              ( if_Hea5867803462524415986on_nat
                                              @ ( ( Minlow != none_nat )
                                                & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                              @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi2 @ Aktnode @ L )
                                                @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                              @ ( heap_T3669509953089699273on_nat @ ( Vebt_predi2 @ Summary2 @ H )
                                                @ ^ [Predsum: option_nat] :
                                                    ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                    @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                      @ ^ [Nextnode: vEBT_VEBTi] :
                                                          ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                          @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info2 )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X3 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                  @ T3 )
                @ Ta
                @ Xa2
                @ R3
                @ N4 )
             => ( P @ T3 @ X3 @ Ta @ Xa2 @ R3 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc8911080112929139129on_nat @ vEBT_vebt_predi @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc6438938002899911481_nat_o @ P @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% vebt_predi.raw_induct
thf(fact_1013_vebt__succi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > option_nat > nat > $o,Xa: produc3881548065746020326Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ! [Vebt_succi2: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: option_nat,N5: nat] :
              ( ( heap_T306965388786959644on_nat @ ( Vebt_succi2 @ A6 @ B5 ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: option_nat,N4: nat] :
              ( ( heap_T306965388786959644on_nat
                @ ( vEBT_c6250501799366334488on_nat
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ^ [Mima: product_prod_nat_nat] :
                          ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X3 @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X3 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [L: nat] :
                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                            @ ^ [Maxlow: option_nat] :
                                                ( if_Hea5867803462524415986on_nat
                                                @ ( ( Maxlow != none_nat )
                                                  & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi2 @ Aktnode @ L )
                                                  @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( Vebt_succi2 @ Summary2 @ H )
                                                  @ ^ [Succsum: option_nat] :
                                                      ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                            @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                      @ Info2 )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X3 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                  @ T3 )
                @ Ta
                @ Xa2
                @ R3
                @ N4 )
             => ( P @ T3 @ X3 @ Ta @ Xa2 @ R3 @ N4 ) ) )
     => ( ( heap_T306965388786959644on_nat @ ( produc8911080112929139129on_nat @ vEBT_vebt_succi @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc6438938002899911481_nat_o @ P @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% vebt_succi.raw_induct
thf(fact_1014_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X2: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg4 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_1015_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg4: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg4 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X2 = Mi )
          | ( X2 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_1016_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary4 ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_1017_delt__out__of__range,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
        | ( ord_less_nat @ Ma @ X2 ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ).

% delt_out_of_range
thf(fact_1018_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T2 @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_1019_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T2 @ zero_zero_nat ) ).

% valid_0_not
thf(fact_1020_deg__deg__n,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( Deg4 = N2 ) ) ).

% deg_deg_n
thf(fact_1021_delete__pres__valid,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) @ N2 ) ) ).

% delete_pres_valid
thf(fact_1022_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_1023_maxbmo,axiom,
    ! [T2: vEBT_VEBT,X2: nat] :
      ( ( ( vEBT_vebt_maxt @ T2 )
        = ( some_nat @ X2 ) )
     => ( vEBT_V8194947554948674370ptions @ T2 @ X2 ) ) ).

% maxbmo
thf(fact_1024_dele__bmo__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_V8194947554948674370ptions @ T2 @ Y2 ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_1025_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
        = ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).

% both_member_options_equiv_member
thf(fact_1026_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
       => ( vEBT_vebt_member @ T2 @ X2 ) ) ) ).

% valid_member_both_member_options
thf(fact_1027_dele__member__cont__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T2 @ X2 ) @ Y2 )
        = ( ( X2 != Y2 )
          & ( vEBT_vebt_member @ T2 @ Y2 ) ) ) ) ).

% dele_member_cont_corr
thf(fact_1028_mint__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).

% mint_member
thf(fact_1029_maxt__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T2 @ Maxi ) ) ) ).

% maxt_member
thf(fact_1030_mint__corr__help,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Mini: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T2 @ X2 )
         => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).

% mint_corr_help
thf(fact_1031_maxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Maxi: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T2 @ X2 )
         => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_1032_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,B2: nat] :
            ( P
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_1033_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A2: int,B2: int] :
            ( P
            & ( Q @ A2 @ B2 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_1034_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X2: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X2 )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_1035_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ( ord_less_eq_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_1036_del__single__cont,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( X2 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ).

% del_single_cont
thf(fact_1037_misiz,axiom,
    ! [T2: vEBT_VEBT,N2: nat,M: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( some_nat @ M )
          = ( vEBT_vebt_mint @ T2 ) )
       => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% misiz
thf(fact_1038_helpyd,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpyd
thf(fact_1039_helpypredd,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some_nat @ Y2 ) )
       => ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% helpypredd
thf(fact_1040_valid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_V8194947554948674370ptions @ T2 @ X2 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y2 ) @ X2 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_1041_valid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X2 ) @ X2 ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_1042_post__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T2 @ X2 ) @ Y2 )
           => ( ( vEBT_vebt_member @ T2 @ Y2 )
              | ( X2 = Y2 ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_1043_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_1044_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 ) ) ) ) ).

% mi_ma_2_deg
thf(fact_1045_member__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_vebt_member @ T2 @ X2 )
        = ( member_nat @ X2 @ ( vEBT_set_vebt @ T2 ) ) ) ) ).

% member_correct
thf(fact_1046_both__member__options__ding,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) @ X2 ) ) ) ) ).

% both_member_options_ding
thf(fact_1047_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_1048_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_1049_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N4: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_1050_Collect__case__prod__mono,axiom,
    ! [A4: nat > nat > $o,B6: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A4 @ B6 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B6 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_1051_Collect__case__prod__mono,axiom,
    ! [A4: int > int > $o,B6: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A4 @ B6 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A4 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B6 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_1052_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu: nat,Uv: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_1053_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu: int,Uv: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_1054_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% in_children_def
thf(fact_1055_maxt__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).

% maxt_corr
thf(fact_1056_maxt__sound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
       => ( ( vEBT_vebt_maxt @ T2 )
          = ( some_nat @ X2 ) ) ) ) ).

% maxt_sound
thf(fact_1057_mint__sound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 )
       => ( ( vEBT_vebt_mint @ T2 )
          = ( some_nat @ X2 ) ) ) ) ).

% mint_sound
thf(fact_1058_mint__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = ( some_nat @ X2 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 ) ) ) ).

% mint_corr
thf(fact_1059_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M4: nat] :
              ( ( ( some_nat @ M4 )
                = ( vEBT_vebt_mint @ Summary4 ) )
              & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_1060_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList2 @ Summary4 ) @ X2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
thf(fact_1061_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_1062_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T ) ) ) ) ).

% set_vebt'_def
thf(fact_1063_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_1064_vebt__succ_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( 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 @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_1065_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S3 ) ) ) ).

% deg_SUcn_Node
thf(fact_1066_set__vebt__set__vebt_H__valid,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_set_vebt @ T2 )
        = ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).

% set_vebt_set_vebt'_valid
thf(fact_1067_inthall,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,N2: nat] :
      ( ! [X3: vEBT_VEBTi] :
          ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1068_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1069_inthall,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o,N2: nat] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
       => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1070_inthall,axiom,
    ! [Xs2: list_real,P: real > $o,N2: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
       => ( P @ ( nth_real @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1071_inthall,axiom,
    ! [Xs2: list_o,P: $o > $o,N2: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1072_inthall,axiom,
    ! [Xs2: list_nat,P: nat > $o,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1073_inthall,axiom,
    ! [Xs2: list_int,P: int > $o,N2: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P @ X3 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% inthall
thf(fact_1074_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1075_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1076_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 )
     => ( ( 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 @ Summary4 @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1077_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_1078_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_1079_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_1080_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_1081_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_1082_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1083_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1084_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1085_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1086_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% diff_numeral_special(9)
thf(fact_1087_diff__numeral__special_I9_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% diff_numeral_special(9)
thf(fact_1088_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1089_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1090_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1091_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_2184487114949457152l_num1 @ zero_z3563351764282998399l_num1 @ ( suc @ N2 ) )
      = zero_z3563351764282998399l_num1 ) ).

% power_0_Suc
thf(fact_1092_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_1093_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_1094_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_1095_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_1096_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_1097_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ ( suc @ N2 ) )
      = zero_z3403309356797280102nteger ) ).

% power_0_Suc
thf(fact_1098_power__Suc0__right,axiom,
    ! [A3: nat] :
      ( ( power_power_nat @ A3 @ ( suc @ zero_zero_nat ) )
      = A3 ) ).

% power_Suc0_right
thf(fact_1099_power__Suc0__right,axiom,
    ! [A3: int] :
      ( ( power_power_int @ A3 @ ( suc @ zero_zero_nat ) )
      = A3 ) ).

% power_Suc0_right
thf(fact_1100_power__Suc0__right,axiom,
    ! [A3: real] :
      ( ( power_power_real @ A3 @ ( suc @ zero_zero_nat ) )
      = A3 ) ).

% power_Suc0_right
thf(fact_1101_power__Suc0__right,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ A3 @ ( suc @ zero_zero_nat ) )
      = A3 ) ).

% power_Suc0_right
thf(fact_1102_power__Suc0__right,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ A3 @ ( suc @ zero_zero_nat ) )
      = A3 ) ).

% power_Suc0_right
thf(fact_1103_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1104_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_1105_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_1106_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_1107_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_1108_nat__power__eq__Suc__0__iff,axiom,
    ! [X2: nat,M: nat] :
      ( ( ( power_power_nat @ X2 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_1109_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_1110_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1111_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1112_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_1113_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_1114_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_1115_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_1116_pred__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T2 @ Y2 )
        & ( ord_less_nat @ Y2 @ X2 )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T2 @ Z3 )
              & ( ord_less_nat @ Z3 @ X2 ) )
           => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ).

% pred_member
thf(fact_1117_succ__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat,Y2: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Y2 )
      = ( ( vEBT_vebt_member @ T2 @ Y2 )
        & ( ord_less_nat @ X2 @ Y2 )
        & ! [Z3: nat] :
            ( ( ( vEBT_vebt_member @ T2 @ Z3 )
              & ( ord_less_nat @ X2 @ Z3 ) )
           => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ).

% succ_member
thf(fact_1118_succ__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% succ_corr
thf(fact_1119_pred__corr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T2 ) @ X2 @ Px ) ) ) ).

% pred_corr
thf(fact_1120_succ__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% succ_correct
thf(fact_1121_pred__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T2 ) @ X2 @ Sx ) ) ) ).

% pred_correct
thf(fact_1122_Suc__inject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% Suc_inject
thf(fact_1123_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_1124_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_1125_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_1126_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_1127_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_1128_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_1129_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1130_diff__Suc__less,axiom,
    ! [N2: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_1131_diff__divide__distrib,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1132_diff__divide__distrib,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ).

% diff_divide_distrib
thf(fact_1133_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_1134_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_1135_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L2 )
       => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_1136_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1137_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1138_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1139_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1140_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).

% diff_le_mono
thf(fact_1141_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_1142_le__diff__iff_H,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ C )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% le_diff_iff'
thf(fact_1143_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).

% diff_le_mono2
thf(fact_1144_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1145_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1146_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1147_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1148_old_Onat_Oexhaust,axiom,
    ! [Y2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y2
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1149_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_1150_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P @ X3 @ Y3 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_1151_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N4: nat] :
            ( ( P @ ( suc @ N4 ) )
           => ( P @ N4 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1152_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1153_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1154_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1155_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M4: nat] :
          ( N2
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_1156_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_1157_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_1158_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_1159_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_1160_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_1161_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_1162_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ N2 )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ I4 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1163_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_1164_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_1165_Nat_OAll__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ N2 )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ I4 ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_1166_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M5: nat] :
            ( ( M
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N2 @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1167_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_1168_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_1169_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_1170_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1171_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1172_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1173_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_1174_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_1175_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_1176_Suc__le__D,axiom,
    ! [N2: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
     => ? [M4: nat] :
          ( M6
          = ( suc @ M4 ) ) ) ).

% Suc_le_D
thf(fact_1177_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_1178_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_1179_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_1180_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N4 )
             => ( P @ M2 ) )
         => ( P @ N4 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_1181_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ M @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_1182_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R4: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X3: nat] : ( R4 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z4: nat] :
              ( ( R4 @ X3 @ Y3 )
             => ( ( R4 @ Y3 @ Z4 )
               => ( R4 @ X3 @ Z4 ) ) )
         => ( ! [N4: nat] : ( R4 @ N4 @ ( suc @ N4 ) )
           => ( R4 @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1183_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1184_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1185_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M3 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_1186_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_1187_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_1188_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_1189_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1190_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1191_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1192_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1193_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ N2 @ N6 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1194_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1195_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1196_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1197_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1198_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1199_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_1200_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ A3 )
       => ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1201_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1202_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1203_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1204_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1205_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N6 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_1206_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1207_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_rat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1208_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_num @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1209_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1210_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N6: nat] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
     => ( ( ord_less_eq_nat @ N2 @ N6 )
       => ( ord_less_eq_int @ ( F @ N6 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_1211_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
            & ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
            & ( P @ ( suc @ I4 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1212_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M3: nat] :
            ( N2
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1213_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ N2 )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1214_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M4: nat] :
          ( N2
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1215_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1216_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1217_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_1218_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_1219_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ N4 )
                 => ( P @ ( suc @ N4 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1220_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P @ ( suc @ N4 ) )
                 => ( P @ N4 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1221_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_1222_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_1223_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_1224_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1225_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_1226_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_1227_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_1228_power__inject__base,axiom,
    ! [A3: real,N2: nat,B3: real] :
      ( ( ( power_power_real @ A3 @ ( suc @ N2 ) )
        = ( power_power_real @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_1229_power__inject__base,axiom,
    ! [A3: code_integer,N2: nat,B3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) )
        = ( power_8256067586552552935nteger @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_1230_power__inject__base,axiom,
    ! [A3: rat,N2: nat,B3: rat] :
      ( ( ( power_power_rat @ A3 @ ( suc @ N2 ) )
        = ( power_power_rat @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_1231_power__inject__base,axiom,
    ! [A3: nat,N2: nat,B3: nat] :
      ( ( ( power_power_nat @ A3 @ ( suc @ N2 ) )
        = ( power_power_nat @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_1232_power__inject__base,axiom,
    ! [A3: int,N2: nat,B3: int] :
      ( ( ( power_power_int @ A3 @ ( suc @ N2 ) )
        = ( power_power_int @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% power_inject_base
thf(fact_1233_power__le__imp__le__base,axiom,
    ! [A3: real,N2: nat,B3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N2 ) ) @ ( power_power_real @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_1234_power__le__imp__le__base,axiom,
    ! [A3: code_integer,N2: nat,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) ) @ ( power_8256067586552552935nteger @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_1235_power__le__imp__le__base,axiom,
    ! [A3: rat,N2: nat,B3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( suc @ N2 ) ) @ ( power_power_rat @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_1236_power__le__imp__le__base,axiom,
    ! [A3: nat,N2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N2 ) ) @ ( power_power_nat @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_1237_power__le__imp__le__base,axiom,
    ! [A3: int,N2: nat,B3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N2 ) ) @ ( power_power_int @ B3 @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% power_le_imp_le_base
thf(fact_1238_power__gt1,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_1239_power__gt1,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ord_less_real @ one_one_real @ ( power_power_real @ A3 @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_1240_power__gt1,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A3 @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_1241_power__gt1,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A3 @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_1242_power__gt1,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ord_less_int @ one_one_int @ ( power_power_int @ A3 @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_1243_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1244_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K2 )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1245_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( P @ N4 )
               => ( P @ ( suc @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1246_power__gt__expt,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).

% power_gt_expt
thf(fact_1247_nat__one__le__power,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_1248_power2__commute,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1249_power2__commute,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1250_power2__commute,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( minus_minus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1251_power2__commute,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1252_power2__commute,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y2 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1253_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_1254_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_1255_power__diff,axiom,
    ! [A3: complex,N2: nat,M: nat] :
      ( ( A3 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_complex @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A3 @ M ) @ ( power_power_complex @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1256_power__diff,axiom,
    ! [A3: code_integer,N2: nat,M: nat] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1257_power__diff,axiom,
    ! [A3: real,N2: nat,M: nat] :
      ( ( A3 != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1258_power__diff,axiom,
    ! [A3: rat,N2: nat,M: nat] :
      ( ( A3 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1259_power__diff,axiom,
    ! [A3: nat,N2: nat,M: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1260_power__diff,axiom,
    ! [A3: int,N2: nat,M: nat] :
      ( ( A3 != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_1261_power__Suc__le__self,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ A3 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ).

% power_Suc_le_self
thf(fact_1262_power__Suc__le__self,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le3102999989581377725nteger @ A3 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ).

% power_Suc_le_self
thf(fact_1263_power__Suc__le__self,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ A3 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ).

% power_Suc_le_self
thf(fact_1264_power__Suc__le__self,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ).

% power_Suc_le_self
thf(fact_1265_power__Suc__le__self,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ A3 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ ( suc @ N2 ) ) @ A3 ) ) ) ).

% power_Suc_le_self
thf(fact_1266_power__Suc__less__one,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le6747313008572928689nteger @ A3 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_1267_power__Suc__less__one,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ A3 @ one_one_real )
       => ( ord_less_real @ ( power_power_real @ A3 @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).

% power_Suc_less_one
thf(fact_1268_power__Suc__less__one,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ A3 @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A3 @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_1269_power__Suc__less__one,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ A3 @ one_one_nat )
       => ( ord_less_nat @ ( power_power_nat @ A3 @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).

% power_Suc_less_one
thf(fact_1270_power__Suc__less__one,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ A3 @ one_one_int )
       => ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).

% power_Suc_less_one
thf(fact_1271_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1272_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_1273_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1274_option_Osize_I3_J,axiom,
    ( ( size_s3991424295186984831it_nat @ none_P1551326421579882414it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1275_option_Osize_I3_J,axiom,
    ( ( size_s364044314319911927it_nat @ none_P7668321371905463026it_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1276_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1277_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1278_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1279_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_1280_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary4 ) @ X2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
thf(fact_1281_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_z3403309356797280102nteger ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1282_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1283_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1284_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_1285_power__diff__power__eq,axiom,
    ! [A3: code_integer,N2: nat,M: nat] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
            = ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1286_power__diff__power__eq,axiom,
    ! [A3: nat,N2: nat,M: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) )
            = ( power_power_nat @ A3 @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1287_power__diff__power__eq,axiom,
    ! [A3: int,N2: nat,M: nat] :
      ( ( A3 != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) )
            = ( power_power_int @ A3 @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_1288_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_1289_less__2__cases,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
     => ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases
thf(fact_1290_less__2__cases__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ( N2 = zero_zero_nat )
        | ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% less_2_cases_iff
thf(fact_1291_power__sub,axiom,
    ! [N2: nat,M: nat,A3: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ A3 )
       => ( ( power_power_nat @ A3 @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) ) ) ) ) ).

% power_sub
thf(fact_1292_power__minus__is__div,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A3 @ B3 ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% power_minus_is_div
thf(fact_1293_div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div_2_gt_zero
thf(fact_1294_Suc__n__div__2__gt__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Suc_n_div_2_gt_zero
thf(fact_1295_less__two__pow__divD,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
        & ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% less_two_pow_divD
thf(fact_1296_less__two__pow__divI,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_nat @ X2 @ ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% less_two_pow_divI
thf(fact_1297_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_1298_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb )
      = none_nat ) ).

% vebt_pred.simps(4)
thf(fact_1299_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) @ Uu2 )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
thf(fact_1300_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_1301_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_1302_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( ( X2 != Mi )
       => ( ( X2 != Ma )
         => ( ~ ( ord_less_nat @ X2 @ Mi )
            & ( ~ ( ord_less_nat @ X2 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X2 )
                & ( ~ ( ord_less_nat @ Ma @ X2 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_1303_Suc__diff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_1304_not__Some__eq2,axiom,
    ! [V: option2661157926820139483um_num] :
      ( ( ! [X: num,Y: num] :
            ( V
           != ( some_P6201964756284913402um_num @ ( product_Pair_num_num @ X @ Y ) ) ) )
      = ( V = none_P4394680061957285238um_num ) ) ).

% not_Some_eq2
thf(fact_1305_not__Some__eq2,axiom,
    ! [V: option254855292876462358T_VEBT] :
      ( ( ! [X: nat,Y: produc4813437837504472865T_VEBT] :
            ( V
           != ( some_P5996745733903548321T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) ) ) )
      = ( V = none_P330983522480640549T_VEBT ) ) ).

% not_Some_eq2
thf(fact_1306_not__Some__eq2,axiom,
    ! [V: option642762832853965969at_num] :
      ( ( ! [X: nat,Y: num] :
            ( V
           != ( some_P8071634352977444016at_num @ ( product_Pair_nat_num @ X @ Y ) ) ) )
      = ( V = none_P6264349658649815852at_num ) ) ).

% not_Some_eq2
thf(fact_1307_not__Some__eq2,axiom,
    ! [V: option4624381673175914239nt_int] :
      ( ( ! [X: int,Y: int] :
            ( V
           != ( some_P4184893108420464158nt_int @ ( product_Pair_int_int @ X @ Y ) ) ) )
      = ( V = none_P2377608414092835994nt_int ) ) ).

% not_Some_eq2
thf(fact_1308_not__Some__eq2,axiom,
    ! [V: option2621746655072343315it_nat] :
      ( ( ! [X: option_nat,Y: produc6653097349344004940it_nat] :
            ( V
           != ( some_P2407035485129114418it_nat @ ( produc61566615109097733it_nat @ X @ Y ) ) ) )
      = ( V = none_P1551326421579882414it_nat ) ) ).

% not_Some_eq2
thf(fact_1309_not__Some__eq2,axiom,
    ! [V: option7339022715339332451it_nat] :
      ( ( ! [X: $o,Y: produc6653097349344004940it_nat] :
            ( V
           != ( some_P468703482102919278it_nat @ ( produc6655106138504972685it_nat @ X @ Y ) ) ) )
      = ( V = none_P7668321371905463026it_nat ) ) ).

% not_Some_eq2
thf(fact_1310_not__Some__eq2,axiom,
    ! [V: option4927543243414619207at_nat] :
      ( ( ! [X: nat,Y: nat] :
            ( V
           != ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ Y ) ) ) )
      = ( V = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq2
thf(fact_1311_vebt__pred_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != none_nat ) ) )
       => ( ! [A: $o] :
              ( ? [Uw2: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A
                     => ( Y2 = none_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B
                       => ( ( A
                           => ( Y2
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A
                           => ( Y2 = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
               => ( Y2 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y2 != none_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y2 != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_1312_vebt__succ_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa )
        = Y2 )
     => ( ! [Uu3: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ Uu3 @ B ) )
           => ( ( Xa = zero_zero_nat )
             => ~ ( ( B
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B
                   => ( Y2 = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2 != none_nat ) ) )
         => ( ( ? [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
             => ( Y2 != none_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y2 != none_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y2 != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_1313_zero__comp__diff__simps_I2_J,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
      = ( ord_less_real @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1314_zero__comp__diff__simps_I2_J,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
      = ( ord_less_rat @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1315_zero__comp__diff__simps_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
      = ( ord_less_int @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(2)
thf(fact_1316_diff__gt__0__iff__gt,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
      = ( ord_less_real @ B3 @ A3 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1317_diff__gt__0__iff__gt,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
      = ( ord_less_rat @ B3 @ A3 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1318_diff__gt__0__iff__gt,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
      = ( ord_less_int @ B3 @ A3 ) ) ).

% diff_gt_0_iff_gt
thf(fact_1319_zero__comp__diff__simps_I1_J,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
      = ( ord_less_eq_real @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1320_zero__comp__diff__simps_I1_J,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
      = ( ord_less_eq_rat @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1321_zero__comp__diff__simps_I1_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
      = ( ord_less_eq_int @ B3 @ A3 ) ) ).

% zero_comp_diff_simps(1)
thf(fact_1322_diff__ge__0__iff__ge,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
      = ( ord_less_eq_real @ B3 @ A3 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1323_diff__ge__0__iff__ge,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A3 @ B3 ) )
      = ( ord_less_eq_rat @ B3 @ A3 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1324_diff__ge__0__iff__ge,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A3 @ B3 ) )
      = ( ord_less_eq_int @ B3 @ A3 ) ) ).

% diff_ge_0_iff_ge
thf(fact_1325_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary4 ) ) ).

% vebt_insert.simps(4)
thf(fact_1326_inrange,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% inrange
thf(fact_1327_Leaf__0__not,axiom,
    ! [A3: $o,B3: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1328_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( N2 = one_one_nat )
       => ? [A: $o,B: $o] :
            ( T2
            = ( vEBT_Leaf @ A @ B ) ) ) ) ).

% deg_1_Leafy
thf(fact_1329_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T2 @ one_one_nat )
     => ? [A: $o,B: $o] :
          ( T2
          = ( vEBT_Leaf @ A @ B ) ) ) ).

% deg_1_Leaf
thf(fact_1330_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T2 @ one_one_nat )
      = ( ? [A2: $o,B2: $o] :
            ( T2
            = ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ).

% deg1Leaf
thf(fact_1331_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_1332_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_1333_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_1334_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_1335_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A3 @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1336_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: real] :
      ( ( minus_minus_real @ A3 @ A3 )
      = zero_zero_real ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1337_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: rat] :
      ( ( minus_minus_rat @ A3 @ A3 )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1338_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ A3 @ A3 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1339_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A3: int] :
      ( ( minus_minus_int @ A3 @ A3 )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1340_diff__zero,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = A3 ) ).

% diff_zero
thf(fact_1341_diff__zero,axiom,
    ! [A3: real] :
      ( ( minus_minus_real @ A3 @ zero_zero_real )
      = A3 ) ).

% diff_zero
thf(fact_1342_diff__zero,axiom,
    ! [A3: rat] :
      ( ( minus_minus_rat @ A3 @ zero_zero_rat )
      = A3 ) ).

% diff_zero
thf(fact_1343_diff__zero,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% diff_zero
thf(fact_1344_diff__zero,axiom,
    ! [A3: int] :
      ( ( minus_minus_int @ A3 @ zero_zero_int )
      = A3 ) ).

% diff_zero
thf(fact_1345_zero__diff,axiom,
    ! [A3: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1346_diff__0__right,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = A3 ) ).

% diff_0_right
thf(fact_1347_diff__0__right,axiom,
    ! [A3: real] :
      ( ( minus_minus_real @ A3 @ zero_zero_real )
      = A3 ) ).

% diff_0_right
thf(fact_1348_diff__0__right,axiom,
    ! [A3: rat] :
      ( ( minus_minus_rat @ A3 @ zero_zero_rat )
      = A3 ) ).

% diff_0_right
thf(fact_1349_diff__0__right,axiom,
    ! [A3: int] :
      ( ( minus_minus_int @ A3 @ zero_zero_int )
      = A3 ) ).

% diff_0_right
thf(fact_1350_diff__self,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ A3 @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% diff_self
thf(fact_1351_diff__self,axiom,
    ! [A3: real] :
      ( ( minus_minus_real @ A3 @ A3 )
      = zero_zero_real ) ).

% diff_self
thf(fact_1352_diff__self,axiom,
    ! [A3: rat] :
      ( ( minus_minus_rat @ A3 @ A3 )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1353_diff__self,axiom,
    ! [A3: int] :
      ( ( minus_minus_int @ A3 @ A3 )
      = zero_zero_int ) ).

% diff_self
thf(fact_1354_vebt__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A3 )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B3 )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1355_vebt__insert_Osimps_I1_J,axiom,
    ! [X2: nat,A3: $o,B3: $o] :
      ( ( ( X2 = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
          = ( vEBT_Leaf @ $true @ B3 ) ) )
      & ( ( X2 != zero_zero_nat )
       => ( ( ( X2 = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
              = ( vEBT_Leaf @ A3 @ $true ) ) )
          & ( ( X2 != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_1356_zero__reorient,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1 = X2 )
      = ( X2 = zero_z3563351764282998399l_num1 ) ) ).

% zero_reorient
thf(fact_1357_zero__reorient,axiom,
    ! [X2: real] :
      ( ( zero_zero_real = X2 )
      = ( X2 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1358_zero__reorient,axiom,
    ! [X2: rat] :
      ( ( zero_zero_rat = X2 )
      = ( X2 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1359_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1360_zero__reorient,axiom,
    ! [X2: int] :
      ( ( zero_zero_int = X2 )
      = ( X2 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1361_ord__eq__le__eq__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C: set_nat,D: set_nat] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C )
       => ( ( C = D )
         => ( ord_less_eq_set_nat @ A3 @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1362_ord__eq__le__eq__trans,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ( C = D )
         => ( ord_less_eq_rat @ A3 @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1363_ord__eq__le__eq__trans,axiom,
    ! [A3: num,B3: num,C: num,D: num] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_num @ B3 @ C )
       => ( ( C = D )
         => ( ord_less_eq_num @ A3 @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1364_ord__eq__le__eq__trans,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ( C = D )
         => ( ord_less_eq_nat @ A3 @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1365_ord__eq__le__eq__trans,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ( C = D )
         => ( ord_less_eq_int @ A3 @ D ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1366_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv3: $o,Uw3: $o,N2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv3 @ Uw3 ) @ ( suc @ N2 ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_1367_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv3: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv3 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_1368_one__reorient,axiom,
    ! [X2: uint32] :
      ( ( one_one_uint32 = X2 )
      = ( X2 = one_one_uint32 ) ) ).

% one_reorient
thf(fact_1369_one__reorient,axiom,
    ! [X2: real] :
      ( ( one_one_real = X2 )
      = ( X2 = one_one_real ) ) ).

% one_reorient
thf(fact_1370_one__reorient,axiom,
    ! [X2: rat] :
      ( ( one_one_rat = X2 )
      = ( X2 = one_one_rat ) ) ).

% one_reorient
thf(fact_1371_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_1372_one__reorient,axiom,
    ! [X2: int] :
      ( ( one_one_int = X2 )
      = ( X2 = one_one_int ) ) ).

% one_reorient
thf(fact_1373_bex2I,axiom,
    ! [A3: num,B3: num,S4: set_Pr8218934625190621173um_num,P: num > num > $o] :
      ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A3 @ B3 ) @ S4 )
     => ( ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A3 @ B3 ) @ S4 )
         => ( P @ A3 @ B3 ) )
       => ? [A: num,B: num] :
            ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ A @ B ) @ S4 )
            & ( P @ A @ B ) ) ) ) ).

% bex2I
thf(fact_1374_bex2I,axiom,
    ! [A3: nat,B3: produc4813437837504472865T_VEBT,S4: set_Pr563407847431865468T_VEBT,P: nat > produc4813437837504472865T_VEBT > $o] :
      ( ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) @ S4 )
     => ( ( ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) @ S4 )
         => ( P @ A3 @ B3 ) )
       => ? [A: nat,B: produc4813437837504472865T_VEBT] :
            ( ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ A @ B ) @ S4 )
            & ( P @ A @ B ) ) ) ) ).

% bex2I
thf(fact_1375_bex2I,axiom,
    ! [A3: nat,B3: num,S4: set_Pr6200539531224447659at_num,P: nat > num > $o] :
      ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A3 @ B3 ) @ S4 )
     => ( ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A3 @ B3 ) @ S4 )
         => ( P @ A3 @ B3 ) )
       => ? [A: nat,B: num] :
            ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ A @ B ) @ S4 )
            & ( P @ A @ B ) ) ) ) ).

% bex2I
thf(fact_1376_bex2I,axiom,
    ! [A3: nat,B3: nat,S4: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S4 )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S4 )
         => ( P @ A3 @ B3 ) )
       => ? [A: nat,B: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S4 )
            & ( P @ A @ B ) ) ) ) ).

% bex2I
thf(fact_1377_bex2I,axiom,
    ! [A3: int,B3: int,S4: set_Pr958786334691620121nt_int,P: int > int > $o] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S4 )
     => ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A3 @ B3 ) @ S4 )
         => ( P @ A3 @ B3 ) )
       => ? [A: int,B: int] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ S4 )
            & ( P @ A @ B ) ) ) ) ).

% bex2I
thf(fact_1378_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N2: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
thf(fact_1379_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
thf(fact_1380_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
thf(fact_1381_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ X @ Y )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ X @ Z3 )
               => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1382_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X: nat,Y: nat] :
          ( ( member_nat @ Y @ Xs )
          & ( ord_less_nat @ Y @ X )
          & ! [Z3: nat] :
              ( ( member_nat @ Z3 @ Xs )
             => ( ( ord_less_nat @ Z3 @ X )
               => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1383_vebt__pred_Osimps_I2_J,axiom,
    ! [A3: $o,Uw3: $o] :
      ( ( A3
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw3 ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A3
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ Uw3 ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_1384_vebt__succ_Osimps_I1_J,axiom,
    ! [B3: $o,Uu2: $o] :
      ( ( B3
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1385_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) )
     => ( ! [A: $o,B: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A: $o,B: $o,N4: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N4 ) ) ) )
         => ( ! [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) @ Uu3 ) )
           => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ X3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
thf(fact_1386_vebt__pred_Osimps_I3_J,axiom,
    ! [B3: $o,A3: $o,Va2: nat] :
      ( ( B3
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( A3
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A3
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_1387_zero__le,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).

% zero_le
thf(fact_1388_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_1389_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1390_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1391_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_1392_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y5 = Z2 )
    = ( ^ [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
          ( ( minus_4019991460397169231l_num1 @ A2 @ B2 )
          = zero_z3563351764282998399l_num1 ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1393_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z2: real] : Y5 = Z2 )
    = ( ^ [A2: real,B2: real] :
          ( ( minus_minus_real @ A2 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1394_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z2: rat] : Y5 = Z2 )
    = ( ^ [A2: rat,B2: rat] :
          ( ( minus_minus_rat @ A2 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1395_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1396_diff__eq__diff__less__eq,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A3 @ B3 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A3 @ B3 )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1397_diff__eq__diff__less__eq,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A3 @ B3 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A3 @ B3 )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1398_diff__eq__diff__less__eq,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A3 @ B3 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A3 @ B3 )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1399_diff__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1400_diff__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1401_diff__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).

% diff_right_mono
thf(fact_1402_diff__left__mono,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1403_diff__left__mono,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A3 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1404_diff__left__mono,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).

% diff_left_mono
thf(fact_1405_diff__mono,axiom,
    ! [A3: real,B3: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1406_diff__mono,axiom,
    ! [A3: rat,B3: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1407_diff__mono,axiom,
    ! [A3: int,B3: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% diff_mono
thf(fact_1408_diff__strict__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1409_diff__strict__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1410_diff__strict__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1411_diff__strict__left__mono,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ord_less_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1412_diff__strict__left__mono,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A3 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1413_diff__strict__left__mono,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_int @ B3 @ A3 )
     => ( ord_less_int @ ( minus_minus_int @ C @ A3 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).

% diff_strict_left_mono
thf(fact_1414_diff__eq__diff__less,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A3 @ B3 )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A3 @ B3 )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1415_diff__eq__diff__less,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A3 @ B3 )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A3 @ B3 )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1416_diff__eq__diff__less,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A3 @ B3 )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A3 @ B3 )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1417_diff__strict__mono,axiom,
    ! [A3: real,B3: real,D: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1418_diff__strict__mono,axiom,
    ! [A3: rat,B3: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A3 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1419_diff__strict__mono,axiom,
    ! [A3: int,B3: int,D: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A3 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1420_exists__leI,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [N7: nat] :
            ( ( ord_less_nat @ N7 @ N2 )
           => ~ ( P @ N7 ) )
       => ( P @ N2 ) )
     => ? [N8: nat] :
          ( ( ord_less_eq_nat @ N8 @ N2 )
          & ( P @ N8 ) ) ) ).

% exists_leI
thf(fact_1421_fstE,axiom,
    ! [X2: product_prod_num_num,A3: num,B3: num,P: num > $o] :
      ( ( X2
        = ( product_Pair_num_num @ A3 @ B3 ) )
     => ( ( P @ ( product_fst_num_num @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1422_fstE,axiom,
    ! [X2: produc8398139464844984134T_VEBT,A3: nat,B3: produc4813437837504472865T_VEBT,P: nat > $o] :
      ( ( X2
        = ( produc1750349459881913976T_VEBT @ A3 @ B3 ) )
     => ( ( P @ ( produc758997459209783180T_VEBT @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1423_fstE,axiom,
    ! [X2: product_prod_nat_num,A3: nat,B3: num,P: nat > $o] :
      ( ( X2
        = ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( ( P @ ( product_fst_nat_num @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1424_fstE,axiom,
    ! [X2: product_prod_nat_nat,A3: nat,B3: nat,P: nat > $o] :
      ( ( X2
        = ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( ( P @ ( product_fst_nat_nat @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1425_fstE,axiom,
    ! [X2: product_prod_int_int,A3: int,B3: int,P: int > $o] :
      ( ( X2
        = ( product_Pair_int_int @ A3 @ B3 ) )
     => ( ( P @ ( product_fst_int_int @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1426_fstE,axiom,
    ! [X2: produc6575502325842934193n_assn,A3: assn,B3: assn,P: assn > $o] :
      ( ( X2
        = ( produc118845697133431529n_assn @ A3 @ B3 ) )
     => ( ( P @ ( produc9167289414957590229n_assn @ X2 ) )
       => ( P @ A3 ) ) ) ).

% fstE
thf(fact_1427_sndE,axiom,
    ! [X2: product_prod_num_num,A3: num,B3: num,P: num > $o] :
      ( ( X2
        = ( product_Pair_num_num @ A3 @ B3 ) )
     => ( ( P @ ( product_snd_num_num @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1428_sndE,axiom,
    ! [X2: produc8398139464844984134T_VEBT,A3: nat,B3: produc4813437837504472865T_VEBT,P: produc4813437837504472865T_VEBT > $o] :
      ( ( X2
        = ( produc1750349459881913976T_VEBT @ A3 @ B3 ) )
     => ( ( P @ ( produc2084898568784432842T_VEBT @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1429_sndE,axiom,
    ! [X2: product_prod_nat_num,A3: nat,B3: num,P: num > $o] :
      ( ( X2
        = ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( ( P @ ( product_snd_nat_num @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1430_sndE,axiom,
    ! [X2: product_prod_nat_nat,A3: nat,B3: nat,P: nat > $o] :
      ( ( X2
        = ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( ( P @ ( product_snd_nat_nat @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1431_sndE,axiom,
    ! [X2: product_prod_int_int,A3: int,B3: int,P: int > $o] :
      ( ( X2
        = ( product_Pair_int_int @ A3 @ B3 ) )
     => ( ( P @ ( product_snd_int_int @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1432_sndE,axiom,
    ! [X2: produc6575502325842934193n_assn,A3: assn,B3: assn,P: assn > $o] :
      ( ( X2
        = ( produc118845697133431529n_assn @ A3 @ B3 ) )
     => ( ( P @ ( produc2051961928117032727n_assn @ X2 ) )
       => ( P @ B3 ) ) ) ).

% sndE
thf(fact_1433_All__prod__contract,axiom,
    ! [P: nat > nat > $o] :
      ( ( ! [A2: nat,X6: nat] : ( P @ A2 @ X6 ) )
      = ( ! [Z3: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Z3 ) @ ( product_snd_nat_nat @ Z3 ) ) ) ) ).

% All_prod_contract
thf(fact_1434_All__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ! [A2: int,X6: int] : ( P @ A2 @ X6 ) )
      = ( ! [Z3: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z3 ) @ ( product_snd_int_int @ Z3 ) ) ) ) ).

% All_prod_contract
thf(fact_1435_All__prod__contract,axiom,
    ! [P: assn > assn > $o] :
      ( ( ! [A2: assn,X6: assn] : ( P @ A2 @ X6 ) )
      = ( ! [Z3: produc6575502325842934193n_assn] : ( P @ ( produc9167289414957590229n_assn @ Z3 ) @ ( produc2051961928117032727n_assn @ Z3 ) ) ) ) ).

% All_prod_contract
thf(fact_1436_Ex__prod__contract,axiom,
    ! [P: nat > nat > $o] :
      ( ( ? [A2: nat,X6: nat] : ( P @ A2 @ X6 ) )
      = ( ? [Z3: product_prod_nat_nat] : ( P @ ( product_fst_nat_nat @ Z3 ) @ ( product_snd_nat_nat @ Z3 ) ) ) ) ).

% Ex_prod_contract
thf(fact_1437_Ex__prod__contract,axiom,
    ! [P: int > int > $o] :
      ( ( ? [A2: int,X6: int] : ( P @ A2 @ X6 ) )
      = ( ? [Z3: product_prod_int_int] : ( P @ ( product_fst_int_int @ Z3 ) @ ( product_snd_int_int @ Z3 ) ) ) ) ).

% Ex_prod_contract
thf(fact_1438_Ex__prod__contract,axiom,
    ! [P: assn > assn > $o] :
      ( ( ? [A2: assn,X6: assn] : ( P @ A2 @ X6 ) )
      = ( ? [Z3: produc6575502325842934193n_assn] : ( P @ ( produc9167289414957590229n_assn @ Z3 ) @ ( produc2051961928117032727n_assn @ Z3 ) ) ) ) ).

% Ex_prod_contract
thf(fact_1439_fn__fst__conv,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X ) ) )
      = ( produc2626176000494625587at_nat
        @ ^ [A2: nat,Uu: nat] : ( F @ A2 ) ) ) ).

% fn_fst_conv
thf(fact_1440_fn__fst__conv,axiom,
    ! [F: nat > $o] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_fst_nat_nat @ X ) ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [A2: nat,Uu: nat] : ( F @ A2 ) ) ) ).

% fn_fst_conv
thf(fact_1441_fn__fst__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,Uu: int] : ( F @ A2 ) ) ) ).

% fn_fst_conv
thf(fact_1442_fn__fst__conv,axiom,
    ! [F: int > $o] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc4947309494688390418_int_o
        @ ^ [A2: int,Uu: int] : ( F @ A2 ) ) ) ).

% fn_fst_conv
thf(fact_1443_fn__fst__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_fst_int_int @ X ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [A2: int,Uu: int] : ( F @ A2 ) ) ) ).

% fn_fst_conv
thf(fact_1444_fn__snd__conv,axiom,
    ! [F: nat > product_prod_nat_nat] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X ) ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Uu: nat] : F ) ) ).

% fn_snd_conv
thf(fact_1445_fn__snd__conv,axiom,
    ! [F: nat > $o] :
      ( ( ^ [X: product_prod_nat_nat] : ( F @ ( product_snd_nat_nat @ X ) ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [Uu: nat] : F ) ) ).

% fn_snd_conv
thf(fact_1446_fn__snd__conv,axiom,
    ! [F: int > product_prod_int_int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Uu: int] : F ) ) ).

% fn_snd_conv
thf(fact_1447_fn__snd__conv,axiom,
    ! [F: int > $o] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc4947309494688390418_int_o
        @ ^ [Uu: int] : F ) ) ).

% fn_snd_conv
thf(fact_1448_fn__snd__conv,axiom,
    ! [F: int > int] :
      ( ( ^ [X: product_prod_int_int] : ( F @ ( product_snd_int_int @ X ) ) )
      = ( produc8211389475949308722nt_int
        @ ^ [Uu: int] : F ) ) ).

% fn_snd_conv
thf(fact_1449_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A2: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1450_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1451_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1452_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A2: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1453_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A2: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A2 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_1454_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_1455_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N4 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct'
thf(fact_1456_nat__compl__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N4 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N4 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_compl_induct
thf(fact_1457_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ord_less_nat @ A3 @ B3 )
        & ( ord_less_eq_nat @ B3 @ ( suc @ A3 ) ) )
      = ( B3
        = ( suc @ A3 ) ) ) ).

% nat_in_between_eq(1)
thf(fact_1458_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ord_less_eq_nat @ A3 @ B3 )
        & ( ord_less_nat @ B3 @ ( suc @ A3 ) ) )
      = ( B3 = A3 ) ) ).

% nat_in_between_eq(2)
thf(fact_1459_Suc__to__right,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( suc @ N2 )
        = M )
     => ( N2
        = ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_1460_nat__geq__1__eq__neqz,axiom,
    ! [X2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X2 )
      = ( X2 != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_1461_obtain__list__from__elements,axiom,
    ! [N2: nat,P: vEBT_VEBT > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: vEBT_VEBT] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_VEBT_VEBT @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1462_obtain__list__from__elements,axiom,
    ! [N2: nat,P: vEBT_VEBTi > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: vEBT_VEBTi] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_VEBT_VEBTi @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1463_obtain__list__from__elements,axiom,
    ! [N2: nat,P: real > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: real] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_real] :
            ( ( ( size_size_list_real @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_real @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1464_obtain__list__from__elements,axiom,
    ! [N2: nat,P: $o > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: $o] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_o] :
            ( ( ( size_size_list_o @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_o @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1465_obtain__list__from__elements,axiom,
    ! [N2: nat,P: nat > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: nat] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_nat] :
            ( ( ( size_size_list_nat @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_nat @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1466_obtain__list__from__elements,axiom,
    ! [N2: nat,P: int > nat > $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ? [Li: int] : ( P @ Li @ I2 ) )
     => ~ ! [L3: list_int] :
            ( ( ( size_size_list_int @ L3 )
              = N2 )
           => ~ ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ N2 )
                 => ( P @ ( nth_int @ L3 @ I3 ) @ I3 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1467_vebt__insert_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info4 @ zero_zero_nat @ Ts @ S ) @ X2 )
      = ( vEBT_Node @ Info4 @ zero_zero_nat @ Ts @ S ) ) ).

% vebt_insert.simps(2)
thf(fact_1468_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) @ X2 ) ).

% vebt_member.simps(2)
thf(fact_1469_le__some__optE,axiom,
    ! [M: set_nat,X2: option_set_nat] :
      ( ( ord_le2843612097646854710et_nat @ ( some_set_nat @ M ) @ X2 )
     => ~ ! [M7: set_nat] :
            ( ( X2
              = ( some_set_nat @ M7 ) )
           => ~ ( ord_less_eq_set_nat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_1470_le__some__optE,axiom,
    ! [M: rat,X2: option_rat] :
      ( ( ord_le2406147912482264968on_rat @ ( some_rat @ M ) @ X2 )
     => ~ ! [M7: rat] :
            ( ( X2
              = ( some_rat @ M7 ) )
           => ~ ( ord_less_eq_rat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_1471_le__some__optE,axiom,
    ! [M: num,X2: option_num] :
      ( ( ord_le6622620407824499402on_num @ ( some_num @ M ) @ X2 )
     => ~ ! [M7: num] :
            ( ( X2
              = ( some_num @ M7 ) )
           => ~ ( ord_less_eq_num @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_1472_le__some__optE,axiom,
    ! [M: nat,X2: option_nat] :
      ( ( ord_le5914376470875661696on_nat @ ( some_nat @ M ) @ X2 )
     => ~ ! [M7: nat] :
            ( ( X2
              = ( some_nat @ M7 ) )
           => ~ ( ord_less_eq_nat @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_1473_le__some__optE,axiom,
    ! [M: int,X2: option_int] :
      ( ( ord_le1736525451366464988on_int @ ( some_int @ M ) @ X2 )
     => ~ ! [M7: int] :
            ( ( X2
              = ( some_int @ M7 ) )
           => ~ ( ord_less_eq_int @ M @ M7 ) ) ) ).

% le_some_optE
thf(fact_1474_vebt__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
             => ~ ( ( Xa != Mi2 )
                 => ( ( Xa != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_1475_vebt__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y2
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => Y2 )
         => ( ( ? [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
             => Y2 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y2 )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                      = ( ~ ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_1476_vebt__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( ( Xa = zero_zero_nat )
               => A )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_1477_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s7982070591426661849_VEBTi @ L2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1478_all__set__conv__nth,axiom,
    ! [L2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ L2 ) )
           => ( P @ ( nth_VEBT_VEBT @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1479_all__set__conv__nth,axiom,
    ! [L2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ L2 ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1480_all__set__conv__nth,axiom,
    ! [L2: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_real @ L2 ) )
           => ( P @ ( nth_real @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1481_all__set__conv__nth,axiom,
    ! [L2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ L2 ) )
           => ( P @ ( nth_o @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1482_all__set__conv__nth,axiom,
    ! [L2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ L2 ) )
           => ( P @ ( nth_nat @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1483_all__set__conv__nth,axiom,
    ! [L2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ L2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ L2 ) )
           => ( P @ ( nth_int @ L2 @ I4 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1484_vebt__insert_Osimps_I3_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info4 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X2 )
      = ( vEBT_Node @ Info4 @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).

% vebt_insert.simps(3)
thf(fact_1485_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X2 ) ).

% vebt_member.simps(3)
thf(fact_1486_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M ) ) ) ).

% nz_le_conv_less
thf(fact_1487_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ one_one_nat @ M )
       => ( ( suc @ ( minus_minus_nat @ N2 @ M ) )
          = ( minus_minus_nat @ N2 @ ( minus_minus_nat @ M @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_1488_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).

% vebt_member.simps(4)
thf(fact_1489_vebt__maxt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( B
                 => ( Y2
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B
                 => ( ( A
                     => ( Y2
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
               => ( Y2
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_1490_vebt__mint_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( A
                 => ( Y2
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A
                 => ( ( B
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B
                     => ( Y2 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
               => ( Y2
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_1491_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_1492_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X122 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X122 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_1493_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_1494_VEBT_Ocase__distrib,axiom,
    ! [H2: produc819165548630102716T_VEBT > produc819165548630102716T_VEBT,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F2: $o > $o > produc819165548630102716T_VEBT,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_c634343235235684882T_VEBT @ F1 @ F2 @ VEBT ) )
      = ( vEBT_c634343235235684882T_VEBT
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_1495_VEBT_Ocase__distrib,axiom,
    ! [H2: produc819165548630102716T_VEBT > $o,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F2: $o > $o > produc819165548630102716T_VEBT,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_c634343235235684882T_VEBT @ F1 @ F2 @ VEBT ) )
      = ( vEBT_case_VEBT_o
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_1496_VEBT_Ocase__distrib,axiom,
    ! [H2: $o > produc819165548630102716T_VEBT,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F2: $o > $o > $o,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_case_VEBT_o @ F1 @ F2 @ VEBT ) )
      = ( vEBT_c634343235235684882T_VEBT
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_1497_VEBT_Ocase__distrib,axiom,
    ! [H2: $o > $o,F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F2: $o > $o > $o,VEBT: vEBT_VEBT] :
      ( ( H2 @ ( vEBT_case_VEBT_o @ F1 @ F2 @ VEBT ) )
      = ( vEBT_case_VEBT_o
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: list_VEBT_VEBT,X42: vEBT_VEBT] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBT ) ) ).

% VEBT.case_distrib
thf(fact_1498_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2574133891255291104it_nat] :
      ( ! [Uu3: produc120671012495760973it_nat > produc120671012495760973it_nat > $o,Uv2: option2621746655072343315it_nat] :
          ( X2
         != ( produc5936680911947247184it_nat @ Uu3 @ ( produc6851560022941992023it_nat @ none_P1551326421579882414it_nat @ Uv2 ) ) )
     => ( ! [Uw2: produc120671012495760973it_nat > produc120671012495760973it_nat > $o,V2: produc120671012495760973it_nat] :
            ( X2
           != ( produc5936680911947247184it_nat @ Uw2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ V2 ) @ none_P1551326421579882414it_nat ) ) )
       => ~ ! [F5: produc120671012495760973it_nat > produc120671012495760973it_nat > $o,X3: produc120671012495760973it_nat,Y3: produc120671012495760973it_nat] :
              ( X2
             != ( produc5936680911947247184it_nat @ F5 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ X3 ) @ ( some_P2407035485129114418it_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc6000686143695694318it_nat] :
      ( ! [Uu3: produc8047831477865546771it_nat > produc8047831477865546771it_nat > $o,Uv2: option7339022715339332451it_nat] :
          ( X2
         != ( produc3920266370798870110it_nat @ Uu3 @ ( produc9206348758962449759it_nat @ none_P7668321371905463026it_nat @ Uv2 ) ) )
     => ( ! [Uw2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > $o,V2: produc8047831477865546771it_nat] :
            ( X2
           != ( produc3920266370798870110it_nat @ Uw2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ V2 ) @ none_P7668321371905463026it_nat ) ) )
       => ~ ! [F5: produc8047831477865546771it_nat > produc8047831477865546771it_nat > $o,X3: produc8047831477865546771it_nat,Y3: produc8047831477865546771it_nat] :
              ( X2
             != ( produc3920266370798870110it_nat @ F5 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ X3 ) @ ( some_P468703482102919278it_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc2233624965454879586on_nat] :
      ( ! [Uu3: nat > nat > $o,Uv2: option_nat] :
          ( X2
         != ( produc4035269172776083154on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X2
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F5: nat > nat > $o,X3: nat,Y3: nat] :
              ( X2
             != ( produc4035269172776083154on_nat @ F5 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc5491161045314408544at_nat] :
      ( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc3994169339658061776at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X2
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F5: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X2
             != ( produc3994169339658061776at_nat @ F5 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1502_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X2: produc7036089656553540234on_num] :
      ( ! [Uu3: num > num > $o,Uv2: option_num] :
          ( X2
         != ( produc3576312749637752826on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X2
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F5: num > num > $o,X3: num,Y3: num] :
              ( X2
             != ( produc3576312749637752826on_num @ F5 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1503_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc6872358179685758443it_nat] :
      ( ! [Uu3: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Uv2: option2621746655072343315it_nat] :
          ( X2
         != ( produc8579712001971957723it_nat @ Uu3 @ ( produc6851560022941992023it_nat @ none_P1551326421579882414it_nat @ Uv2 ) ) )
     => ( ! [Uw2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,V2: produc120671012495760973it_nat] :
            ( X2
           != ( produc8579712001971957723it_nat @ Uw2 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ V2 ) @ none_P1551326421579882414it_nat ) ) )
       => ~ ! [F5: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,A: produc120671012495760973it_nat,B: produc120671012495760973it_nat] :
              ( X2
             != ( produc8579712001971957723it_nat @ F5 @ ( produc6851560022941992023it_nat @ ( some_P2407035485129114418it_nat @ A ) @ ( some_P2407035485129114418it_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1504_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5059602919146741221it_nat] :
      ( ! [Uu3: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Uv2: option7339022715339332451it_nat] :
          ( X2
         != ( produc2320005133921938071it_nat @ Uu3 @ ( produc9206348758962449759it_nat @ none_P7668321371905463026it_nat @ Uv2 ) ) )
     => ( ! [Uw2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,V2: produc8047831477865546771it_nat] :
            ( X2
           != ( produc2320005133921938071it_nat @ Uw2 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ V2 ) @ none_P7668321371905463026it_nat ) ) )
       => ~ ! [F5: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,A: produc8047831477865546771it_nat,B: produc8047831477865546771it_nat] :
              ( X2
             != ( produc2320005133921938071it_nat @ F5 @ ( produc9206348758962449759it_nat @ ( some_P468703482102919278it_nat @ A ) @ ( some_P468703482102919278it_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1505_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc8306885398267862888on_nat] :
      ( ! [Uu3: nat > nat > nat,Uv2: option_nat] :
          ( X2
         != ( produc8929957630744042906on_nat @ Uu3 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X2
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F5: nat > nat > nat,A: nat,B: nat] :
              ( X2
             != ( produc8929957630744042906on_nat @ F5 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1506_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc5542196010084753463at_nat] :
      ( ! [Uu3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X2
         != ( produc2899441246263362727at_nat @ Uu3 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X2
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
              ( X2
             != ( produc2899441246263362727at_nat @ F5 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1507_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X2: produc1193250871479095198on_num] :
      ( ! [Uu3: num > num > num,Uv2: option_num] :
          ( X2
         != ( produc5778274026573060048on_num @ Uu3 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X2
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F5: num > num > num,A: num,B: num] :
              ( X2
             != ( produc5778274026573060048on_num @ F5 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1508_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_1509_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_1510_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_1511_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_1512_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_1513_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu3: $o,Uv2: $o,D2: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ D2 ) )
     => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg5: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) @ Deg5 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_1514_VEBT_Oexhaust,axiom,
    ! [Y2: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X123: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y2
         != ( vEBT_Node @ X112 @ X123 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y2
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_1515_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X122 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_1516_VEBT_Odisc_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] : ( vEBT_is_Node @ ( vEBT_Node @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBT.disc(1)
thf(fact_1517_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( VEBT
        = ( vEBT_Node @ X11 @ X122 @ X13 @ X14 ) )
     => ( vEBT_is_Node @ VEBT ) ) ).

% VEBT.discI(1)
thf(fact_1518_is__Node__def,axiom,
    ( vEBT_is_Node
    = ( ^ [VEBT2: vEBT_VEBT] :
        ? [X113: option4927543243414619207at_nat,X124: nat,X133: list_VEBT_VEBT,X143: vEBT_VEBT] :
          ( VEBT2
          = ( vEBT_Node @ X113 @ X124 @ X133 @ X143 ) ) ) ) ).

% is_Node_def
thf(fact_1519_VEBT_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F2: $o > $o > produc819165548630102716T_VEBT,X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_c634343235235684882T_VEBT @ F1 @ F2 @ ( vEBT_Node @ X11 @ X122 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBT.simps(5)
thf(fact_1520_VEBT_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F2: $o > $o > $o,X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_case_VEBT_o @ F1 @ F2 @ ( vEBT_Node @ X11 @ X122 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBT.simps(5)
thf(fact_1521_VEBT_Odisc_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ~ ( vEBT_is_Node @ ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.disc(2)
thf(fact_1522_VEBT_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > produc819165548630102716T_VEBT,F2: $o > $o > produc819165548630102716T_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_c634343235235684882T_VEBT @ F1 @ F2 @ ( vEBT_Leaf @ X21 @ X222 ) )
      = ( F2 @ X21 @ X222 ) ) ).

% VEBT.simps(6)
thf(fact_1523_VEBT_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > $o,F2: $o > $o > $o,X21: $o,X222: $o] :
      ( ( vEBT_case_VEBT_o @ F1 @ F2 @ ( vEBT_Leaf @ X21 @ X222 ) )
      = ( F2 @ X21 @ X222 ) ) ).

% VEBT.simps(6)
thf(fact_1524_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A3 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1525_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A3: num,B3: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A3 ) @ ( some_num @ B3 ) )
      = ( some_num @ ( F @ A3 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1526_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A3: nat,B3: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) )
      = ( some_nat @ ( F @ A3 @ B3 ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1527_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Uv3: option2621746655072343315it_nat] :
      ( ( vEBT_V819568868292977612it_nat @ Uu2 @ none_P1551326421579882414it_nat @ Uv3 )
      = none_P1551326421579882414it_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1528_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Uv3: option7339022715339332451it_nat] :
      ( ( vEBT_V613753007643960916it_nat @ Uu2 @ none_P7668321371905463026it_nat @ Uv3 )
      = none_P7668321371905463026it_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1529_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv3: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu2 @ none_P5556105721700978146at_nat @ Uv3 )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1530_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: num > num > num,Uv3: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu2 @ none_num @ Uv3 )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1531_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: nat > nat > nat,Uv3: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu2 @ none_nat @ Uv3 )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1532_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T ) ) ) ) ).

% set_vebt_def
thf(fact_1533_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw3: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,V: produc120671012495760973it_nat] :
      ( ( vEBT_V819568868292977612it_nat @ Uw3 @ ( some_P2407035485129114418it_nat @ V ) @ none_P1551326421579882414it_nat )
      = none_P1551326421579882414it_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1534_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw3: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,V: produc8047831477865546771it_nat] :
      ( ( vEBT_V613753007643960916it_nat @ Uw3 @ ( some_P468703482102919278it_nat @ V ) @ none_P7668321371905463026it_nat )
      = none_P7668321371905463026it_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1535_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw3 @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1536_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw3: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw3 @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1537_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw3: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw3 @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1538_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: produc120671012495760973it_nat > produc120671012495760973it_nat > produc120671012495760973it_nat,Xa: option2621746655072343315it_nat,Xb: option2621746655072343315it_nat,Y2: option2621746655072343315it_nat] :
      ( ( ( vEBT_V819568868292977612it_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = none_P1551326421579882414it_nat )
         => ( Y2 != none_P1551326421579882414it_nat ) )
       => ( ( ? [V2: produc120671012495760973it_nat] :
                ( Xa
                = ( some_P2407035485129114418it_nat @ V2 ) )
           => ( ( Xb = none_P1551326421579882414it_nat )
             => ( Y2 != none_P1551326421579882414it_nat ) ) )
         => ~ ! [A: produc120671012495760973it_nat] :
                ( ( Xa
                  = ( some_P2407035485129114418it_nat @ A ) )
               => ! [B: produc120671012495760973it_nat] :
                    ( ( Xb
                      = ( some_P2407035485129114418it_nat @ B ) )
                   => ( Y2
                     != ( some_P2407035485129114418it_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1539_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: produc8047831477865546771it_nat > produc8047831477865546771it_nat > produc8047831477865546771it_nat,Xa: option7339022715339332451it_nat,Xb: option7339022715339332451it_nat,Y2: option7339022715339332451it_nat] :
      ( ( ( vEBT_V613753007643960916it_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = none_P7668321371905463026it_nat )
         => ( Y2 != none_P7668321371905463026it_nat ) )
       => ( ( ? [V2: produc8047831477865546771it_nat] :
                ( Xa
                = ( some_P468703482102919278it_nat @ V2 ) )
           => ( ( Xb = none_P7668321371905463026it_nat )
             => ( Y2 != none_P7668321371905463026it_nat ) ) )
         => ~ ! [A: produc8047831477865546771it_nat] :
                ( ( Xa
                  = ( some_P468703482102919278it_nat @ A ) )
               => ! [B: produc8047831477865546771it_nat] :
                    ( ( Xb
                      = ( some_P468703482102919278it_nat @ B ) )
                   => ( Y2
                     != ( some_P468703482102919278it_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1540_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y2: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = none_P5556105721700978146at_nat )
         => ( Y2 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y2 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A: product_prod_nat_nat] :
                ( ( Xa
                  = ( some_P7363390416028606310at_nat @ A ) )
               => ! [B: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B ) )
                   => ( Y2
                     != ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1541_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: num > num > num,Xa: option_num,Xb: option_num,Y2: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = none_num )
         => ( Y2 != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y2 != none_num ) ) )
         => ~ ! [A: num] :
                ( ( Xa
                  = ( some_num @ A ) )
               => ! [B: num] :
                    ( ( Xb
                      = ( some_num @ B ) )
                   => ( Y2
                     != ( some_num @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1542_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X2: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y2: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa @ Xb )
        = Y2 )
     => ( ( ( Xa = none_nat )
         => ( Y2 != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y2 != none_nat ) ) )
         => ~ ! [A: nat] :
                ( ( Xa
                  = ( some_nat @ A ) )
               => ! [B: nat] :
                    ( ( Xb
                      = ( some_nat @ B ) )
                   => ( Y2
                     != ( some_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1543_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X3 ) )
     => ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux3 ) )
       => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_1544_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_1545_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_1546_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( X2
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu3: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu3 @ $true ) )
         => ( ! [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
           => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X2
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
thf(fact_1547_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1548_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1549_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( vEBT_Leaf @ A @ B ) )
     => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
thf(fact_1550_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1551_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1552_vebt__mint_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( A3
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A3
       => ( ( B3
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B3
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A3 @ B3 ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_1553_vebt__maxt_Osimps_I1_J,axiom,
    ! [B3: $o,A3: $o] :
      ( ( B3
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B3
       => ( ( A3
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A3
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A3 @ B3 ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_1554_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu3: $o,Uv2: $o,Uw2: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT,Uz3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) @ Uz3 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X3 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_1555_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu3: $o,Uv2: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A: $o,Uw2: $o] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A: $o,B: $o,Va3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT,Vb2: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) @ Vb2 ) )
           => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X2
                 != ( 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] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                      ( X2
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ X3 ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
thf(fact_1556_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [Uu3: $o,B: $o] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ B ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
       => ( ! [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT,Va4: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) @ Va4 ) )
         => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X2
               != ( 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] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                    ( X2
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ X3 ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
thf(fact_1557_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X3 ) )
     => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ X3 ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X3 ) )
         => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
thf(fact_1558_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [X2: produc9072475918466114483BT_nat] :
      ( ! [A: $o,B: $o,X3: nat] :
          ( X2
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X3 ) )
     => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
            ( X2
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ X3 ) )
       => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT,X3: nat] :
              ( X2
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) @ X3 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X2
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
                  ( X2
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ X3 ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
thf(fact_1559_atLeastatMost__subset__iff,axiom,
    ! [A3: set_nat,B3: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A3 @ B3 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A3 @ B3 )
        | ( ( ord_less_eq_set_nat @ C @ A3 )
          & ( ord_less_eq_set_nat @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1560_atLeastatMost__subset__iff,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A3 @ B3 )
        | ( ( ord_less_eq_rat @ C @ A3 )
          & ( ord_less_eq_rat @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1561_atLeastatMost__subset__iff,axiom,
    ! [A3: num,B3: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A3 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A3 @ B3 )
        | ( ( ord_less_eq_num @ C @ A3 )
          & ( ord_less_eq_num @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1562_atLeastatMost__subset__iff,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A3 @ B3 )
        | ( ( ord_less_eq_nat @ C @ A3 )
          & ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1563_atLeastatMost__subset__iff,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A3 @ B3 )
        | ( ( ord_less_eq_int @ C @ A3 )
          & ( ord_less_eq_int @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1564_atLeastatMost__subset__iff,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A3 @ B3 )
        | ( ( ord_less_eq_real @ C @ A3 )
          & ( ord_less_eq_real @ B3 @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_1565_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_1566_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A: $o,B: $o] :
              ( A1
              = ( vEBT_Leaf @ A @ B ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
             => ( ( A22 = Deg )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X4 @ N4 ) )
                 => ( ( vEBT_invar_vebt @ Summary @ M4 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                     => ( ( M4 = N4 )
                       => ( ( Deg
                            = ( plus_plus_nat @ N4 @ M4 ) )
                         => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
               => ( ( A22 = Deg )
                 => ( ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X4 @ N4 ) )
                   => ( ( vEBT_invar_vebt @ Summary @ M4 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                       => ( ( M4
                            = ( suc @ N4 ) )
                         => ( ( Deg
                              = ( plus_plus_nat @ N4 @ M4 ) )
                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
                 => ( ( A22 = Deg )
                   => ( ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X4 @ N4 ) )
                     => ( ( vEBT_invar_vebt @ Summary @ M4 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                         => ( ( M4 = N4 )
                           => ( ( Deg
                                = ( plus_plus_nat @ N4 @ M4 ) )
                             => ( ! [I3: nat] :
                                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                               => ( ( ( 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 ) ) @ Deg ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                    = I3 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                & ! [X4: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                                                        = I3 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X4 )
                                                      & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary: vEBT_VEBT,M4: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg @ TreeList3 @ Summary ) )
                   => ( ( A22 = Deg )
                     => ( ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X4 @ N4 ) )
                       => ( ( vEBT_invar_vebt @ Summary @ M4 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                           => ( ( M4
                                = ( suc @ N4 ) )
                             => ( ( Deg
                                  = ( plus_plus_nat @ N4 @ M4 ) )
                               => ( ! [I3: nat] :
                                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                     => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                                 => ( ( ( 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 ) ) @ Deg ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
                                                      = I3 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
                                                  & ! [X4: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                                                          = I3 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X4 )
                                                        & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_1567_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A2: $o,B2: $o] :
                ( A12
                = ( vEBT_Leaf @ A2 @ B2 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A23
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A23
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A23
                = ( plus_plus_nat @ N @ N ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A23
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_1568_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Uu3: $o,Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ Uu3 @ Uv2 ) )
       => ( ! [Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
               => ( ( Xa = Mi2 )
                  | ( Xa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( ( Xa = Mi2 )
                    | ( Xa = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_1569_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => Y2 )
       => ( ( ? [Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) )
           => Y2 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
               => ( Y2
                  = ( ~ ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y2
                    = ( ~ ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                   => ( Y2
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_1570_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( ( Xa = zero_zero_nat )
               => A )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_1571_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_1572_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y2
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y2 )
         => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y2
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_1573_even__odd__cases,axiom,
    ! [X2: nat] :
      ( ! [N4: nat] :
          ( X2
         != ( plus_plus_nat @ N4 @ N4 ) )
     => ~ ! [N4: nat] :
            ( X2
           != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).

% even_odd_cases
thf(fact_1574_add__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% add_shift
thf(fact_1575_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T @ X )
          | ( vEBT_VEBT_membermima @ T @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1576_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1577_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X2 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
          | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1578_pow__sum,axiom,
    ! [A3: nat,B3: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ B3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ).

% pow_sum
thf(fact_1579_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_1580_add__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = A3 ) ).

% add_0
thf(fact_1581_add__0,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ zero_zero_real @ A3 )
      = A3 ) ).

% add_0
thf(fact_1582_add__0,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A3 )
      = A3 ) ).

% add_0
thf(fact_1583_add__0,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% add_0
thf(fact_1584_add__0,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ zero_zero_int @ A3 )
      = A3 ) ).

% add_0
thf(fact_1585_zero__eq__add__iff__both__eq__0,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X2 @ Y2 ) )
      = ( ( X2 = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1586_add__eq__0__iff__both__eq__0,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( plus_plus_nat @ X2 @ Y2 )
        = zero_zero_nat )
      = ( ( X2 = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1587_add__cancel__right__right,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( A3
        = ( plus_p361126936061061375l_num1 @ A3 @ B3 ) )
      = ( B3 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_right
thf(fact_1588_add__cancel__right__right,axiom,
    ! [A3: real,B3: real] :
      ( ( A3
        = ( plus_plus_real @ A3 @ B3 ) )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_1589_add__cancel__right__right,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3
        = ( plus_plus_rat @ A3 @ B3 ) )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_1590_add__cancel__right__right,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3
        = ( plus_plus_nat @ A3 @ B3 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_1591_add__cancel__right__right,axiom,
    ! [A3: int,B3: int] :
      ( ( A3
        = ( plus_plus_int @ A3 @ B3 ) )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_1592_add__cancel__right__left,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( A3
        = ( plus_p361126936061061375l_num1 @ B3 @ A3 ) )
      = ( B3 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_right_left
thf(fact_1593_add__cancel__right__left,axiom,
    ! [A3: real,B3: real] :
      ( ( A3
        = ( plus_plus_real @ B3 @ A3 ) )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_1594_add__cancel__right__left,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3
        = ( plus_plus_rat @ B3 @ A3 ) )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_1595_add__cancel__right__left,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3
        = ( plus_plus_nat @ B3 @ A3 ) )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_1596_add__cancel__right__left,axiom,
    ! [A3: int,B3: int] :
      ( ( A3
        = ( plus_plus_int @ B3 @ A3 ) )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_1597_add__cancel__left__right,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_right
thf(fact_1598_add__cancel__left__right,axiom,
    ! [A3: real,B3: real] :
      ( ( ( plus_plus_real @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_1599_add__cancel__left__right,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_1600_add__cancel__left__right,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( plus_plus_nat @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_1601_add__cancel__left__right,axiom,
    ! [A3: int,B3: int] :
      ( ( ( plus_plus_int @ A3 @ B3 )
        = A3 )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_1602_add__cancel__left__left,axiom,
    ! [B3: word_N3645301735248828278l_num1,A3: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_z3563351764282998399l_num1 ) ) ).

% add_cancel_left_left
thf(fact_1603_add__cancel__left__left,axiom,
    ! [B3: real,A3: real] :
      ( ( ( plus_plus_real @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_1604_add__cancel__left__left,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ( plus_plus_rat @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_1605_add__cancel__left__left,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ( plus_plus_nat @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_1606_add__cancel__left__left,axiom,
    ! [B3: int,A3: int] :
      ( ( ( plus_plus_int @ B3 @ A3 )
        = A3 )
      = ( B3 = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_1607_double__zero__sym,axiom,
    ! [A3: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A3 @ A3 ) )
      = ( A3 = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1608_double__zero__sym,axiom,
    ! [A3: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A3 @ A3 ) )
      = ( A3 = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1609_double__zero__sym,axiom,
    ! [A3: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A3 @ A3 ) )
      = ( A3 = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1610_add_Oright__neutral,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = A3 ) ).

% add.right_neutral
thf(fact_1611_add_Oright__neutral,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ A3 @ zero_zero_real )
      = A3 ) ).

% add.right_neutral
thf(fact_1612_add_Oright__neutral,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ A3 @ zero_zero_rat )
      = A3 ) ).

% add.right_neutral
thf(fact_1613_add_Oright__neutral,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% add.right_neutral
thf(fact_1614_add_Oright__neutral,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ A3 @ zero_zero_int )
      = A3 ) ).

% add.right_neutral
thf(fact_1615_double__eq__0__iff,axiom,
    ! [A3: real] :
      ( ( ( plus_plus_real @ A3 @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% double_eq_0_iff
thf(fact_1616_double__eq__0__iff,axiom,
    ! [A3: rat] :
      ( ( ( plus_plus_rat @ A3 @ A3 )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_1617_double__eq__0__iff,axiom,
    ! [A3: int] :
      ( ( ( plus_plus_int @ A3 @ A3 )
        = zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% double_eq_0_iff
thf(fact_1618_add__le__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
      = ( ord_less_eq_real @ A3 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_1619_add__le__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_1620_add__le__cancel__right,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_1621_add__le__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
      = ( ord_less_eq_int @ A3 @ B3 ) ) ).

% add_le_cancel_right
thf(fact_1622_add__le__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
      = ( ord_less_eq_real @ A3 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_1623_add__le__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
      = ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_1624_add__le__cancel__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
      = ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_1625_add__le__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
      = ( ord_less_eq_int @ A3 @ B3 ) ) ).

% add_le_cancel_left
thf(fact_1626_add__less__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
      = ( ord_less_real @ A3 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_1627_add__less__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( ord_less_rat @ A3 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_1628_add__less__cancel__right,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_1629_add__less__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
      = ( ord_less_int @ A3 @ B3 ) ) ).

% add_less_cancel_right
thf(fact_1630_add__less__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
      = ( ord_less_real @ A3 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_1631_add__less__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
      = ( ord_less_rat @ A3 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_1632_add__less__cancel__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
      = ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_1633_add__less__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
      = ( ord_less_int @ A3 @ B3 ) ) ).

% add_less_cancel_left
thf(fact_1634_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1635_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1636_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1637_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1638_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1639_add__numeral__left,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Z ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_1640_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_1641_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_1642_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_1643_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_1644_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_1645_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_1646_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1647_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_1648_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_1649_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_1650_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_1651_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_1652_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_1653_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_1654_Icc__eq__Icc,axiom,
    ! [L2: set_nat,H2: set_nat,L4: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
        = ( set_or4548717258645045905et_nat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_set_nat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1655_Icc__eq__Icc,axiom,
    ! [L2: rat,H2: rat,L4: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
        = ( set_or633870826150836451st_rat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
          & ~ ( ord_less_eq_rat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1656_Icc__eq__Icc,axiom,
    ! [L2: num,H2: num,L4: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
        = ( set_or7049704709247886629st_num @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L2 @ H2 )
          & ~ ( ord_less_eq_num @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1657_Icc__eq__Icc,axiom,
    ! [L2: nat,H2: nat,L4: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
        = ( set_or1269000886237332187st_nat @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
          & ~ ( ord_less_eq_nat @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1658_Icc__eq__Icc,axiom,
    ! [L2: int,H2: int,L4: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
        = ( set_or1266510415728281911st_int @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_int @ L2 @ H2 )
          & ~ ( ord_less_eq_int @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1659_Icc__eq__Icc,axiom,
    ! [L2: real,H2: real,L4: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
        = ( set_or1222579329274155063t_real @ L4 @ H3 ) )
      = ( ( ( L2 = L4 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_real @ L2 @ H2 )
          & ~ ( ord_less_eq_real @ L4 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_1660_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1661_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_1662_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1663_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ A3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1664_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_1665_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1666_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ A3 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1667_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
      = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_1668_le__add__same__cancel2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_1669_le__add__same__cancel2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_1670_le__add__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_1671_le__add__same__cancel2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).

% le_add_same_cancel2
thf(fact_1672_le__add__same__cancel1,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_1673_le__add__same__cancel1,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_1674_le__add__same__cancel1,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_1675_le__add__same__cancel1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
      = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).

% le_add_same_cancel1
thf(fact_1676_add__le__same__cancel2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_1677_add__le__same__cancel2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_1678_add__le__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_1679_add__le__same__cancel2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
      = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_1680_add__le__same__cancel1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_1681_add__le__same__cancel1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B3 @ A3 ) @ B3 )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_1682_add__le__same__cancel1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
      = ( ord_less_eq_nat @ A3 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_1683_add__le__same__cancel1,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
      = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_1684_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ A3 ) )
      = ( ord_less_real @ zero_zero_real @ A3 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1685_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ A3 ) )
      = ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1686_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ A3 ) )
      = ( ord_less_int @ zero_zero_int @ A3 ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_1687_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A3 @ A3 ) @ zero_zero_real )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1688_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A3 @ A3 ) @ zero_zero_rat )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1689_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A3 @ A3 ) @ zero_zero_int )
      = ( ord_less_int @ A3 @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_1690_less__add__same__cancel2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
      = ( ord_less_real @ zero_zero_real @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_1691_less__add__same__cancel2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
      = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_1692_less__add__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_1693_less__add__same__cancel2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
      = ( ord_less_int @ zero_zero_int @ B3 ) ) ).

% less_add_same_cancel2
thf(fact_1694_less__add__same__cancel1,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
      = ( ord_less_real @ zero_zero_real @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_1695_less__add__same__cancel1,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
      = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_1696_less__add__same__cancel1,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_1697_less__add__same__cancel1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
      = ( ord_less_int @ zero_zero_int @ B3 ) ) ).

% less_add_same_cancel1
thf(fact_1698_add__less__same__cancel2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ B3 )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_1699_add__less__same__cancel2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ B3 )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_1700_add__less__same__cancel2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_1701_add__less__same__cancel2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
      = ( ord_less_int @ A3 @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_1702_add__less__same__cancel1,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B3 @ A3 ) @ B3 )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_1703_add__less__same__cancel1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B3 @ A3 ) @ B3 )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_1704_add__less__same__cancel1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
      = ( ord_less_nat @ A3 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_1705_add__less__same__cancel1,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
      = ( ord_less_int @ A3 @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_1706_diff__add__zero,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1707_le__add__diff__inverse,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1708_le__add__diff__inverse,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A3 @ B3 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1709_le__add__diff__inverse,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1710_le__add__diff__inverse,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A3 @ B3 ) )
        = A3 ) ) ).

% le_add_diff_inverse
thf(fact_1711_le__add__diff__inverse2,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( plus_plus_real @ ( minus_minus_real @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1712_le__add__diff__inverse2,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1713_le__add__diff__inverse2,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1714_le__add__diff__inverse2,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( plus_plus_int @ ( minus_minus_int @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% le_add_diff_inverse2
thf(fact_1715_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_1716_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_1717_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_1718_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_1719_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ N2 ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1720_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1721_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1722_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1723_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1724_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_1725_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) @ one_one_uint32 )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1726_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ one_on7727431528512463931l_num1 )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1727_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
      = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1728_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1729_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1730_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
      = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_1731_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_1732_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_1733_one__add__one,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ one_one_uint32 )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1734_one__add__one,axiom,
    ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1735_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1736_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1737_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1738_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_1739_add__2__eq__Suc_H,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc'
thf(fact_1740_add__2__eq__Suc,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( suc @ ( suc @ N2 ) ) ) ).

% add_2_eq_Suc
thf(fact_1741_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_1742_sum__power2__eq__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1743_sum__power2__eq__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1744_sum__power2__eq__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1745_sum__power2__eq__zero__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( X2 = zero_z3403309356797280102nteger )
        & ( Y2 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_1746_is__num__normalize_I1_J,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
      = ( plus_plus_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1747_is__num__normalize_I1_J,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
      = ( plus_plus_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1748_is__num__normalize_I1_J,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1749_add_Ogroup__left__neutral,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = A3 ) ).

% add.group_left_neutral
thf(fact_1750_add_Ogroup__left__neutral,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ zero_zero_real @ A3 )
      = A3 ) ).

% add.group_left_neutral
thf(fact_1751_add_Ogroup__left__neutral,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A3 )
      = A3 ) ).

% add.group_left_neutral
thf(fact_1752_add_Ogroup__left__neutral,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ zero_zero_int @ A3 )
      = A3 ) ).

% add.group_left_neutral
thf(fact_1753_add_Ocomm__neutral,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = A3 ) ).

% add.comm_neutral
thf(fact_1754_add_Ocomm__neutral,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ A3 @ zero_zero_real )
      = A3 ) ).

% add.comm_neutral
thf(fact_1755_add_Ocomm__neutral,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ A3 @ zero_zero_rat )
      = A3 ) ).

% add.comm_neutral
thf(fact_1756_add_Ocomm__neutral,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% add.comm_neutral
thf(fact_1757_add_Ocomm__neutral,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ A3 @ zero_zero_int )
      = A3 ) ).

% add.comm_neutral
thf(fact_1758_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_1759_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ zero_zero_real @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_1760_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_1761_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_1762_comm__monoid__add__class_Oadd__0,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ zero_zero_int @ A3 )
      = A3 ) ).

% comm_monoid_add_class.add_0
thf(fact_1763_add__le__imp__le__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
     => ( ord_less_eq_real @ A3 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_1764_add__le__imp__le__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
     => ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_1765_add__le__imp__le__right,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
     => ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_1766_add__le__imp__le__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
     => ( ord_less_eq_int @ A3 @ B3 ) ) ).

% add_le_imp_le_right
thf(fact_1767_add__le__imp__le__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
     => ( ord_less_eq_real @ A3 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_1768_add__le__imp__le__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
     => ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_1769_add__le__imp__le__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
     => ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_1770_add__le__imp__le__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
     => ( ord_less_eq_int @ A3 @ B3 ) ) ).

% add_le_imp_le_left
thf(fact_1771_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C3: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_1772_add__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_1773_add__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_1774_add__right__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_1775_add__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add_right_mono
thf(fact_1776_less__eqE,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ~ ! [C2: nat] :
            ( B3
           != ( plus_plus_nat @ A3 @ C2 ) ) ) ).

% less_eqE
thf(fact_1777_add__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_1778_add__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_1779_add__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_1780_add__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).

% add_left_mono
thf(fact_1781_add__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_1782_add__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_1783_add__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_1784_add__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_mono
thf(fact_1785_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_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_add__less__imp__less__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) )
     => ( ord_less_real @ A3 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_1798_add__less__imp__less__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
     => ( ord_less_rat @ A3 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_1799_add__less__imp__less__right,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
     => ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_1800_add__less__imp__less__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) )
     => ( ord_less_int @ A3 @ B3 ) ) ).

% add_less_imp_less_right
thf(fact_1801_add__less__imp__less__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) )
     => ( ord_less_real @ A3 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_1802_add__less__imp__less__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) )
     => ( ord_less_rat @ A3 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_1803_add__less__imp__less__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) )
     => ( ord_less_nat @ A3 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_1804_add__less__imp__less__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) )
     => ( ord_less_int @ A3 @ B3 ) ) ).

% add_less_imp_less_left
thf(fact_1805_add__strict__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1806_add__strict__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1807_add__strict__right__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1808_add__strict__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1809_add__strict__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ C @ A3 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_1810_add__strict__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A3 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_1811_add__strict__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A3 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_1812_add__strict__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ C @ A3 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).

% add_strict_left_mono
thf(fact_1813_add__strict__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1814_add__strict__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1815_add__strict__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1816_add__strict__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1817_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_1818_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_1819_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_1820_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_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_add__divide__distrib,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A3 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1830_add__divide__distrib,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A3 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ).

% add_divide_distrib
thf(fact_1831_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_1832_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_1833_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A3: nat] :
      ( ( A4
        = ( plus_plus_nat @ K @ A3 ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).

% nat_arith.suc1
thf(fact_1834_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1835_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1836_less__add__eq__less,axiom,
    ! [K: nat,L2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L2 )
     => ( ( ( plus_plus_nat @ M @ L2 )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_1837_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_1838_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_1839_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_1840_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1841_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1842_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_1843_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_1844_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M3: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1845_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_1846_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_1847_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_1848_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_1849_le__Suc__ex,axiom,
    ! [K: nat,L2: nat] :
      ( ( ord_less_eq_nat @ K @ L2 )
     => ? [N4: nat] :
          ( L2
          = ( plus_plus_nat @ K @ N4 ) ) ) ).

% le_Suc_ex
thf(fact_1850_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_1851_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_1852_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_1853_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_1854_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_1855_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_1856_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_1857_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_1858_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_1859_add__nonpos__eq__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X2 @ Y2 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1860_add__nonpos__eq__0__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y2 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X2 @ Y2 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1861_add__nonpos__eq__0__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X2 @ Y2 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1862_add__nonpos__eq__0__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X2 @ Y2 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1863_add__nonneg__eq__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( plus_plus_real @ X2 @ Y2 )
            = zero_zero_real )
          = ( ( X2 = zero_zero_real )
            & ( Y2 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1864_add__nonneg__eq__0__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ( plus_plus_rat @ X2 @ Y2 )
            = zero_zero_rat )
          = ( ( X2 = zero_zero_rat )
            & ( Y2 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1865_add__nonneg__eq__0__iff,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
       => ( ( ( plus_plus_nat @ X2 @ Y2 )
            = zero_zero_nat )
          = ( ( X2 = zero_zero_nat )
            & ( Y2 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1866_add__nonneg__eq__0__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ( plus_plus_int @ X2 @ Y2 )
            = zero_zero_int )
          = ( ( X2 = zero_zero_int )
            & ( Y2 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1867_add__nonpos__nonpos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_1868_add__nonpos__nonpos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1869_add__nonpos__nonpos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_1870_add__nonpos__nonpos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_1871_add__nonneg__nonneg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1872_add__nonneg__nonneg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1873_add__nonneg__nonneg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1874_add__nonneg__nonneg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_1875_add__increasing2,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B3 @ A3 )
       => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1876_add__increasing2,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B3 @ A3 )
       => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1877_add__increasing2,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B3 @ A3 )
       => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1878_add__increasing2,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B3 @ A3 )
       => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).

% add_increasing2
thf(fact_1879_add__decreasing2,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A3 @ B3 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_1880_add__decreasing2,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A3 @ B3 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_1881_add__decreasing2,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_1882_add__decreasing2,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A3 @ B3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing2
thf(fact_1883_add__increasing,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).

% add_increasing
thf(fact_1884_add__increasing,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).

% add_increasing
thf(fact_1885_add__increasing,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).

% add_increasing
thf(fact_1886_add__increasing,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).

% add_increasing
thf(fact_1887_add__decreasing,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B3 )
       => ( ord_less_eq_real @ ( plus_plus_real @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_1888_add__decreasing,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B3 )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_1889_add__decreasing,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B3 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_1890_add__decreasing,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B3 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A3 @ C ) @ B3 ) ) ) ).

% add_decreasing
thf(fact_1891_pos__add__strict,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1892_pos__add__strict,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1893_pos__add__strict,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1894_pos__add__strict,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_1895_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ~ ! [C2: nat] :
            ( ( B3
              = ( plus_plus_nat @ A3 @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1896_add__pos__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_1897_add__pos__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_1898_add__pos__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_1899_add__pos__pos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).

% add_pos_pos
thf(fact_1900_add__neg__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_1901_add__neg__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_1902_add__neg__neg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_1903_add__neg__neg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_1904_add__less__zeroD,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
        | ( ord_less_real @ Y2 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1905_add__less__zeroD,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X2 @ zero_zero_rat )
        | ( ord_less_rat @ Y2 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1906_add__less__zeroD,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ zero_zero_int )
     => ( ( ord_less_int @ X2 @ zero_zero_int )
        | ( ord_less_int @ Y2 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1907_add__less__le__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1908_add__less__le__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1909_add__less__le__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1910_add__less__le__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1911_add__le__less__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1912_add__le__less__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1913_add__le__less__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1914_add__le__less__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1915_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_1916_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_1917_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_1918_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_1919_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_1920_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_1921_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_1922_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_1923_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit0 @ N2 ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ ( numera7442385471795722001l_num1 @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1924_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1925_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1926_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1927_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1928_add__mono1,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( plus_plus_real @ A3 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).

% add_mono1
thf(fact_1929_add__mono1,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( plus_plus_rat @ A3 @ one_one_rat ) @ ( plus_plus_rat @ B3 @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_1930_add__mono1,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_1931_add__mono1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ord_less_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).

% add_mono1
thf(fact_1932_less__add__one,axiom,
    ! [A3: real] : ( ord_less_real @ A3 @ ( plus_plus_real @ A3 @ one_one_real ) ) ).

% less_add_one
thf(fact_1933_less__add__one,axiom,
    ! [A3: rat] : ( ord_less_rat @ A3 @ ( plus_plus_rat @ A3 @ one_one_rat ) ) ).

% less_add_one
thf(fact_1934_less__add__one,axiom,
    ! [A3: nat] : ( ord_less_nat @ A3 @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ).

% less_add_one
thf(fact_1935_less__add__one,axiom,
    ! [A3: int] : ( ord_less_int @ A3 @ ( plus_plus_int @ A3 @ one_one_int ) ) ).

% less_add_one
thf(fact_1936_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ X2 ) )
      = ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ X2 ) @ one_one_uint32 ) ) ).

% one_plus_numeral_commute
thf(fact_1937_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ X2 ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ X2 ) @ one_on7727431528512463931l_num1 ) ) ).

% one_plus_numeral_commute
thf(fact_1938_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X2 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_1939_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_1940_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_1941_one__plus__numeral__commute,axiom,
    ! [X2: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X2 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_1942_diff__le__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
      = ( ord_less_eq_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_1943_diff__le__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
      = ( ord_less_eq_rat @ A3 @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_1944_diff__le__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
      = ( ord_less_eq_int @ A3 @ ( plus_plus_int @ C @ B3 ) ) ) ).

% diff_le_eq
thf(fact_1945_le__diff__eq,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_1946_le__diff__eq,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( minus_minus_rat @ C @ B3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_1947_le__diff__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).

% le_diff_eq
thf(fact_1948_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1949_le__add__diff,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).

% le_add_diff
thf(fact_1950_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1951_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1952_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A3 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1953_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1954_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A3 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A3 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1955_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A3 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A3 ) @ B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1956_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ A3 @ ( minus_minus_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1957_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ A3 @ B3 )
       => ( ( ( minus_minus_nat @ B3 @ A3 )
            = C )
          = ( B3
            = ( plus_plus_nat @ C @ A3 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1958_add__le__imp__le__diff,axiom,
    ! [I: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1959_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1960_add__le__imp__le__diff,axiom,
    ! [I: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1961_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_1962_add__le__add__imp__diff__le,axiom,
    ! [I: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
         => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1963_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1964_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1965_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
         => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_1966_less__diff__eq,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ A3 @ ( minus_minus_real @ C @ B3 ) )
      = ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_1967_less__diff__eq,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ ( minus_minus_rat @ C @ B3 ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_1968_less__diff__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
      = ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).

% less_diff_eq
thf(fact_1969_diff__less__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
      = ( ord_less_real @ A3 @ ( plus_plus_real @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_1970_diff__less__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
      = ( ord_less_rat @ A3 @ ( plus_plus_rat @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_1971_diff__less__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
      = ( ord_less_int @ A3 @ ( plus_plus_int @ C @ B3 ) ) ) ).

% diff_less_eq
thf(fact_1972_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A3: real,B3: real] :
      ( ~ ( ord_less_real @ A3 @ B3 )
     => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A3 @ B3 ) )
        = A3 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1973_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A3: rat,B3: rat] :
      ( ~ ( ord_less_rat @ A3 @ B3 )
     => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A3 @ B3 ) )
        = A3 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1974_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A3: nat,B3: nat] :
      ( ~ ( ord_less_nat @ A3 @ B3 )
     => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A3 @ B3 ) )
        = A3 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1975_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A3: int,B3: int] :
      ( ~ ( ord_less_int @ A3 @ B3 )
     => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A3 @ B3 ) )
        = A3 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1976_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1977_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1978_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q3: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1979_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_1980_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_1981_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M3: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1982_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1983_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1984_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1985_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1986_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1987_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M4: nat,N4: nat] :
          ( ( ord_less_nat @ M4 @ N4 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1988_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1989_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_1990_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_1991_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_1992_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_1993_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_1994_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_1995_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_1996_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit0 @ N2 ) )
      = ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ ( numera7442385471795722001l_num1 @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_1997_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_1998_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_1999_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2000_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_2001_VEBT_Odisc__eq__case_I1_J,axiom,
    ( vEBT_is_Node
    = ( vEBT_case_VEBT_o
      @ ^ [Uu: option4927543243414619207at_nat,Uv: nat,Uw: list_VEBT_VEBT,Ux: vEBT_VEBT] : $true
      @ ^ [Uu: $o,Uv: $o] : $false ) ) ).

% VEBT.disc_eq_case(1)
thf(fact_2002_add__strict__increasing2,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2003_add__strict__increasing2,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2004_add__strict__increasing2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2005_add__strict__increasing2,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_2006_add__strict__increasing,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ B3 @ C )
       => ( ord_less_real @ B3 @ ( plus_plus_real @ A3 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2007_add__strict__increasing,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ B3 @ C )
       => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A3 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2008_add__strict__increasing,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ B3 @ C )
       => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A3 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2009_add__strict__increasing,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ B3 @ C )
       => ( ord_less_int @ B3 @ ( plus_plus_int @ A3 @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_2010_add__pos__nonneg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2011_add__pos__nonneg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2012_add__pos__nonneg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2013_add__pos__nonneg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).

% add_pos_nonneg
thf(fact_2014_add__nonpos__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_2015_add__nonpos__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_2016_add__nonpos__neg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_2017_add__nonpos__neg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_2018_add__nonneg__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A3 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2019_add__nonneg__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A3 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2020_add__nonneg__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2021_add__nonneg__pos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).

% add_nonneg_pos
thf(fact_2022_add__neg__nonpos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_2023_add__neg__nonpos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_2024_add__neg__nonpos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_2025_add__neg__nonpos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_2026_field__le__epsilon,axiom,
    ! [X2: real,Y2: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y2 @ E ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2027_field__le__epsilon,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y2 @ E ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_epsilon
thf(fact_2028_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2029_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2030_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2031_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2032_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2033_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2034_div__add__self2,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2035_div__add__self2,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2036_div__add__self1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
        = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2037_div__add__self1,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2038_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT,Ux2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv3 @ Uw3 ) @ Ux2 ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_2039_less__half__sum,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ A3 @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_2040_less__half__sum,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ A3 @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_2041_gt__half__sum,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B3 ) ) ).

% gt_half_sum
thf(fact_2042_gt__half__sum,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B3 ) ) ).

% gt_half_sum
thf(fact_2043_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A3: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
      = ( ~ ( ( ( ord_less_nat @ A3 @ B3 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A3
                  = ( plus_plus_nat @ B3 @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2044_nat__diff__split,axiom,
    ! [P: nat > $o,A3: nat,B3: nat] :
      ( ( P @ ( minus_minus_nat @ A3 @ B3 ) )
      = ( ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A3
              = ( plus_plus_nat @ B3 @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_2045_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_2046_bounded__Max__nat,axiom,
    ! [P: nat > $o,X2: nat,M8: nat] :
      ( ( P @ X2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M8 ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X4: nat] :
                    ( ( P @ X4 )
                   => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_2047_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc5169207103205389656T_VEBT] :
      ~ ! [F5: nat > produc4813437837504472865T_VEBT > produc4813437837504472865T_VEBT,A: nat,B: nat,Acc: produc4813437837504472865T_VEBT] :
          ( X2
         != ( produc6510890545428481418T_VEBT @ F5 @ ( produc5776433622635646767T_VEBT @ A @ ( produc1750349459881913976T_VEBT @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2048_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc3368934014287244435at_num] :
      ~ ! [F5: nat > num > num,A: nat,B: nat,Acc: num] :
          ( X2
         != ( produc851828971589881931at_num @ F5 @ ( produc1195630363706982562at_num @ A @ ( product_Pair_nat_num @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2049_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X2: produc4471711990508489141at_nat] :
      ~ ! [F5: nat > nat > nat,A: nat,B: nat,Acc: nat] :
          ( X2
         != ( produc3209952032786966637at_nat @ F5 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2050_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_2051_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_2052_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_2053_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = ( ( ( X2 = zero_zero_nat )
         => A3 )
        & ( ( X2 != zero_zero_nat )
         => ( ( ( X2 = one_one_nat )
             => B3 )
            & ( X2 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_2054_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_2055_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2056_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2057_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2058_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2059_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3403309356797280102nteger )
     => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
       != zero_z3403309356797280102nteger ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2060_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_z3563351764282998399l_num1 )
     => ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M )
       != zero_z3563351764282998399l_num1 ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2061_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2062_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2063_div__exp__eq,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2064_div__exp__eq,axiom,
    ! [A3: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( divide1791077408188789448l_num1 @ ( divide1791077408188789448l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide1791077408188789448l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2065_div__exp__eq,axiom,
    ! [A3: uint32,M: nat,N2: nat] :
      ( ( divide_divide_uint32 @ ( divide_divide_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2066_div__exp__eq,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2067_div__exp__eq,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2068_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va2 @ Vb ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_2069_sum__power2__le__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2070_sum__power2__le__zero__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger )
      = ( ( X2 = zero_z3403309356797280102nteger )
        & ( Y2 = zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2071_sum__power2__le__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2072_sum__power2__le__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2073_sum__power2__ge__zero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2074_sum__power2__ge__zero,axiom,
    ! [X2: code_integer,Y2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2075_sum__power2__ge__zero,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2076_sum__power2__ge__zero,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2077_sum__power2__gt__zero__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_z3403309356797280102nteger )
        | ( Y2 != zero_z3403309356797280102nteger ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2078_sum__power2__gt__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2079_sum__power2__gt__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2080_sum__power2__gt__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2081_not__sum__power2__lt__zero,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_z3403309356797280102nteger ) ).

% not_sum_power2_lt_zero
thf(fact_2082_not__sum__power2__lt__zero,axiom,
    ! [X2: real,Y2: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2083_not__sum__power2__lt__zero,axiom,
    ! [X2: rat,Y2: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2084_not__sum__power2__lt__zero,axiom,
    ! [X2: int,Y2: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2085_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N4: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B3 @ N4 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_2086_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N4: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N4 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_2087_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_2088_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_2089_atLeastatMost__psubset__iff,axiom,
    ! [A3: set_nat,B3: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A3 @ B3 ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A3 @ B3 )
          | ( ( ord_less_eq_set_nat @ C @ A3 )
            & ( ord_less_eq_set_nat @ B3 @ D )
            & ( ( ord_less_set_nat @ C @ A3 )
              | ( ord_less_set_nat @ B3 @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2090_atLeastatMost__psubset__iff,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A3 @ B3 )
          | ( ( ord_less_eq_rat @ C @ A3 )
            & ( ord_less_eq_rat @ B3 @ D )
            & ( ( ord_less_rat @ C @ A3 )
              | ( ord_less_rat @ B3 @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2091_atLeastatMost__psubset__iff,axiom,
    ! [A3: num,B3: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A3 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A3 @ B3 )
          | ( ( ord_less_eq_num @ C @ A3 )
            & ( ord_less_eq_num @ B3 @ D )
            & ( ( ord_less_num @ C @ A3 )
              | ( ord_less_num @ B3 @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2092_atLeastatMost__psubset__iff,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A3 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A3 @ B3 )
          | ( ( ord_less_eq_nat @ C @ A3 )
            & ( ord_less_eq_nat @ B3 @ D )
            & ( ( ord_less_nat @ C @ A3 )
              | ( ord_less_nat @ B3 @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2093_atLeastatMost__psubset__iff,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A3 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A3 @ B3 )
          | ( ( ord_less_eq_int @ C @ A3 )
            & ( ord_less_eq_int @ B3 @ D )
            & ( ( ord_less_int @ C @ A3 )
              | ( ord_less_int @ B3 @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2094_atLeastatMost__psubset__iff,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A3 @ B3 )
          | ( ( ord_less_eq_real @ C @ A3 )
            & ( ord_less_eq_real @ B3 @ D )
            & ( ( ord_less_real @ C @ A3 )
              | ( ord_less_real @ B3 @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_2095_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,M: nat,Deg4: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary4 @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg4
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_2096_vebt__delete_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) )
      = ( vEBT_Leaf @ A3 @ B3 ) ) ).

% vebt_delete.simps(3)
thf(fact_2097_vebt__delete_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B3 ) ) ).

% vebt_delete.simps(1)
thf(fact_2098_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) @ Uu2 )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) ) ).

% vebt_delete.simps(4)
thf(fact_2099_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,M: nat,Deg4: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary4 @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg4
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X_1 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_2100_vebt__delete_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A3 @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_2101_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_2102_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S ) @ X2 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_2103_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X2 )
      = ( ( X2 = Mi )
        | ( X2 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_2104_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,M: nat,Deg4: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary4 @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg4
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary4 @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_2105_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,M: nat,Deg4: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary4 @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg4
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary4 @ I2 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 ) )
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I2 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N2 )
                                        = I2 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ Deg4 ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_2106_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X2 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_2107_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
           => ~ ( ( Xa = Mi2 )
                | ( Xa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
             => ~ ( ( Xa = Mi2 )
                  | ( Xa = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_2108_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat,Va2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ( N2
          = ( suc @ ( suc @ Va2 ) ) )
       => ( ~ ( ord_less_nat @ Ma @ Mi )
         => ( ( Ma != Mi )
           => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_2109_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ B ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y2 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y2 = one_one_nat ) )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( Y2
                              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( if_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
thf(fact_2110_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ B ) )
             => ( ? [Va3: nat] :
                    ( Xa
                    = ( suc @ ( suc @ Va3 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y2 = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                             => ( Y2
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
thf(fact_2111_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2 != one_one_nat ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y2 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                     != ( plus_plus_nat @ one_one_nat
                        @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                          @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                            @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Mi2 @ Xa )
                                    & ( ord_less_nat @ Xa @ Ma2 ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                  @ zero_zero_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
thf(fact_2112_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ X2 @ Mi )
       => ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
thf(fact_2113_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Ma: nat,X2: nat,Mi: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ Ma @ X2 )
       => ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
            @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
thf(fact_2114_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( X2 = Mi ) @ zero_zero_nat
          @ ( if_nat @ ( X2 = Ma ) @ zero_zero_nat
            @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ zero_zero_nat
              @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ zero_zero_nat
                @ ( if_nat
                  @ ( ( ord_less_nat @ Mi @ X2 )
                    & ( ord_less_nat @ X2 @ Ma ) )
                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                  @ zero_zero_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
thf(fact_2115_nat__induct2,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct2
thf(fact_2116_field__less__half__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2117_field__less__half__sum,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ X2 @ Y2 )
     => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2118_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L: nat,D3: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D3 ) ) @ L ) ) ) ).

% bit_concat_def
thf(fact_2119_mul__shift,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = Z )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) )
        = ( some_nat @ Z ) ) ) ).

% mul_shift
thf(fact_2120_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_2121_high__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X2 ) @ N2 )
        = Y2 ) ) ).

% high_inv
thf(fact_2122_low__inv,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X2 ) @ N2 )
        = X2 ) ) ).

% low_inv
thf(fact_2123_mult__zero__left,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% mult_zero_left
thf(fact_2124_mult__zero__left,axiom,
    ! [A3: real] :
      ( ( times_times_real @ zero_zero_real @ A3 )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_2125_mult__zero__left,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A3 )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_2126_mult__zero__left,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2127_mult__zero__left,axiom,
    ! [A3: int] :
      ( ( times_times_int @ zero_zero_int @ A3 )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2128_mult__zero__right,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% mult_zero_right
thf(fact_2129_mult__zero__right,axiom,
    ! [A3: real] :
      ( ( times_times_real @ A3 @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_2130_mult__zero__right,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ A3 @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_2131_mult__zero__right,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ A3 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2132_mult__zero__right,axiom,
    ! [A3: int] :
      ( ( times_times_int @ A3 @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2133_mult__eq__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( times_times_real @ A3 @ B3 )
        = zero_zero_real )
      = ( ( A3 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_2134_mult__eq__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( times_times_rat @ A3 @ B3 )
        = zero_zero_rat )
      = ( ( A3 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_2135_mult__eq__0__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( times_times_nat @ A3 @ B3 )
        = zero_zero_nat )
      = ( ( A3 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_2136_mult__eq__0__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ( times_times_int @ A3 @ B3 )
        = zero_zero_int )
      = ( ( A3 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_2137_mult__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( times_times_real @ C @ A3 )
        = ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_2138_mult__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ( times_times_rat @ C @ A3 )
        = ( times_times_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_2139_mult__cancel__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ( times_times_nat @ C @ A3 )
        = ( times_times_nat @ C @ B3 ) )
      = ( ( C = zero_zero_nat )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_2140_mult__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ( times_times_int @ C @ A3 )
        = ( times_times_int @ C @ B3 ) )
      = ( ( C = zero_zero_int )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_left
thf(fact_2141_mult__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ( times_times_real @ A3 @ C )
        = ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_2142_mult__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ( times_times_rat @ A3 @ C )
        = ( times_times_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_2143_mult__cancel__right,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ( times_times_nat @ A3 @ C )
        = ( times_times_nat @ B3 @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_2144_mult__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ( times_times_int @ A3 @ C )
        = ( times_times_int @ B3 @ C ) )
      = ( ( C = zero_zero_int )
        | ( A3 = B3 ) ) ) ).

% mult_cancel_right
thf(fact_2145_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Z ) )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2146_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_2147_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_2148_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_2149_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_2150_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2151_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2152_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2153_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2154_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2155_mult_Oright__neutral,axiom,
    ! [A3: uint32] :
      ( ( times_times_uint32 @ A3 @ one_one_uint32 )
      = A3 ) ).

% mult.right_neutral
thf(fact_2156_mult_Oright__neutral,axiom,
    ! [A3: real] :
      ( ( times_times_real @ A3 @ one_one_real )
      = A3 ) ).

% mult.right_neutral
thf(fact_2157_mult_Oright__neutral,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ A3 @ one_one_rat )
      = A3 ) ).

% mult.right_neutral
thf(fact_2158_mult_Oright__neutral,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ A3 @ one_one_nat )
      = A3 ) ).

% mult.right_neutral
thf(fact_2159_mult_Oright__neutral,axiom,
    ! [A3: int] :
      ( ( times_times_int @ A3 @ one_one_int )
      = A3 ) ).

% mult.right_neutral
thf(fact_2160_mult__1,axiom,
    ! [A3: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A3 )
      = A3 ) ).

% mult_1
thf(fact_2161_mult__1,axiom,
    ! [A3: real] :
      ( ( times_times_real @ one_one_real @ A3 )
      = A3 ) ).

% mult_1
thf(fact_2162_mult__1,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ one_one_rat @ A3 )
      = A3 ) ).

% mult_1
thf(fact_2163_mult__1,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ one_one_nat @ A3 )
      = A3 ) ).

% mult_1
thf(fact_2164_mult__1,axiom,
    ! [A3: int] :
      ( ( times_times_int @ one_one_int @ A3 )
      = A3 ) ).

% mult_1
thf(fact_2165_times__divide__eq__left,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( times_times_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
      = ( divide_divide_real @ ( times_times_real @ B3 @ A3 ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2166_times__divide__eq__left,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
      = ( divide_divide_rat @ ( times_times_rat @ B3 @ A3 ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2167_divide__divide__eq__left,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A3 @ B3 ) @ C )
      = ( divide_divide_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2168_divide__divide__eq__left,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A3 @ B3 ) @ C )
      = ( divide_divide_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2169_divide__divide__eq__right,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( divide_divide_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ).

% divide_divide_eq_right
thf(fact_2170_divide__divide__eq__right,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( divide_divide_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ).

% divide_divide_eq_right
thf(fact_2171_times__divide__eq__right,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2172_times__divide__eq__right,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A3 @ B3 ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2173_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2174_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2175_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2176_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2177_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2178_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2179_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_2180_sum__squares__eq__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
        = zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2181_sum__squares__eq__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) )
        = zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2182_sum__squares__eq__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
        = zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2183_mult__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( C
        = ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2184_mult__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C
        = ( times_times_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_2185_mult__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( C
        = ( times_times_int @ C @ B3 ) )
      = ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2186_mult__cancel__left2,axiom,
    ! [C: real,A3: real] :
      ( ( ( times_times_real @ C @ A3 )
        = C )
      = ( ( C = zero_zero_real )
        | ( A3 = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_2187_mult__cancel__left2,axiom,
    ! [C: rat,A3: rat] :
      ( ( ( times_times_rat @ C @ A3 )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A3 = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_2188_mult__cancel__left2,axiom,
    ! [C: int,A3: int] :
      ( ( ( times_times_int @ C @ A3 )
        = C )
      = ( ( C = zero_zero_int )
        | ( A3 = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_2189_mult__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( C
        = ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( B3 = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2190_mult__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( C
        = ( times_times_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B3 = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_2191_mult__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( C
        = ( times_times_int @ B3 @ C ) )
      = ( ( C = zero_zero_int )
        | ( B3 = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2192_mult__cancel__right2,axiom,
    ! [A3: real,C: real] :
      ( ( ( times_times_real @ A3 @ C )
        = C )
      = ( ( C = zero_zero_real )
        | ( A3 = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_2193_mult__cancel__right2,axiom,
    ! [A3: rat,C: rat] :
      ( ( ( times_times_rat @ A3 @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A3 = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_2194_mult__cancel__right2,axiom,
    ! [A3: int,C: int] :
      ( ( ( times_times_int @ A3 @ C )
        = C )
      = ( ( C = zero_zero_int )
        | ( A3 = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_2195_distrib__right__numeral,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B3 @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2196_distrib__right__numeral,axiom,
    ! [A3: real,B3: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B3 @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2197_distrib__right__numeral,axiom,
    ! [A3: rat,B3: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B3 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2198_distrib__right__numeral,axiom,
    ! [A3: nat,B3: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A3 @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B3 @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2199_distrib__right__numeral,axiom,
    ! [A3: int,B3: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A3 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B3 @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2200_distrib__left__numeral,axiom,
    ! [V: num,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( plus_p361126936061061375l_num1 @ B3 @ C ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B3 ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2201_distrib__left__numeral,axiom,
    ! [V: num,B3: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B3 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B3 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2202_distrib__left__numeral,axiom,
    ! [V: num,B3: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B3 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B3 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2203_distrib__left__numeral,axiom,
    ! [V: num,B3: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B3 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2204_distrib__left__numeral,axiom,
    ! [V: num,B3: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B3 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2205_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
          = ( divide_divide_real @ A3 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2206_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
          = ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2207_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2208_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
        = ( divide_divide_rat @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2209_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2210_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ B3 @ C ) )
        = ( divide_divide_rat @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2211_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2212_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
        = ( divide_divide_rat @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2213_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ C @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2214_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ C @ B3 ) )
        = ( divide_divide_rat @ A3 @ B3 ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2215_div__mult__mult1,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
        = ( divide_divide_nat @ A3 @ B3 ) ) ) ).

% div_mult_mult1
thf(fact_2216_div__mult__mult1,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
        = ( divide_divide_int @ A3 @ B3 ) ) ) ).

% div_mult_mult1
thf(fact_2217_div__mult__mult2,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
        = ( divide_divide_nat @ A3 @ B3 ) ) ) ).

% div_mult_mult2
thf(fact_2218_div__mult__mult2,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ A3 @ B3 ) ) ) ).

% div_mult_mult2
thf(fact_2219_div__mult__mult1__if,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
          = ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2220_div__mult__mult1__if,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
          = ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2221_nonzero__mult__div__cancel__left,axiom,
    ! [A3: real,B3: real] :
      ( ( A3 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ A3 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2222_nonzero__mult__div__cancel__left,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A3 @ B3 ) @ A3 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2223_nonzero__mult__div__cancel__left,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ A3 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2224_nonzero__mult__div__cancel__left,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ A3 )
        = B3 ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2225_nonzero__mult__div__cancel__right,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2226_nonzero__mult__div__cancel__right,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2227_nonzero__mult__div__cancel__right,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2228_nonzero__mult__div__cancel__right,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ B3 )
        = A3 ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2229_right__diff__distrib__numeral,axiom,
    ! [V: num,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( minus_4019991460397169231l_num1 @ B3 @ C ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ B3 ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2230_right__diff__distrib__numeral,axiom,
    ! [V: num,B3: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B3 ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2231_right__diff__distrib__numeral,axiom,
    ! [V: num,B3: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B3 ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2232_right__diff__distrib__numeral,axiom,
    ! [V: num,B3: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2233_left__diff__distrib__numeral,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,V: num] :
      ( ( times_7065122842183080059l_num1 @ ( minus_4019991460397169231l_num1 @ A3 @ B3 ) @ ( numera7442385471795722001l_num1 @ V ) )
      = ( minus_4019991460397169231l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ B3 @ ( numera7442385471795722001l_num1 @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2234_left__diff__distrib__numeral,axiom,
    ! [A3: real,B3: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B3 @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2235_left__diff__distrib__numeral,axiom,
    ! [A3: rat,B3: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B3 @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2236_left__diff__distrib__numeral,axiom,
    ! [A3: int,B3: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A3 @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B3 @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2237_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2238_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2239_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2240_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_2241_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2242_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_2243_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_2244_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_2245_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_2246_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) )
        = A3 )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B3
            = ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2247_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) )
        = A3 )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2248_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( A3
        = ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) )
            = B3 ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2249_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( A3
        = ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) )
            = B3 ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2250_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) ) @ A3 )
      = ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2251_divide__le__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) ) @ A3 )
      = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2252_le__divide__eq__numeral1_I1_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) @ B3 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2253_le__divide__eq__numeral1_I1_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) ) @ B3 ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2254_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) ) @ A3 )
      = ( ord_less_real @ B3 @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2255_divide__less__eq__numeral1_I1_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) ) @ A3 )
      = ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2256_less__divide__eq__numeral1_I1_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) @ B3 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2257_less__divide__eq__numeral1_I1_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A3 @ ( numeral_numeral_rat @ W ) ) @ B3 ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2258_div__mult__self1,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_mult_self1
thf(fact_2259_div__mult__self1,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C @ B3 ) ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_mult_self1
thf(fact_2260_div__mult__self2,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_mult_self2
thf(fact_2261_div__mult__self2,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B3 @ C ) ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_mult_self2
thf(fact_2262_div__mult__self3,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A3 ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_mult_self3
thf(fact_2263_div__mult__self3,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A3 ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_mult_self3
thf(fact_2264_div__mult__self4,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A3 ) @ B3 )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_mult_self4
thf(fact_2265_div__mult__self4,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A3 ) @ B3 )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_mult_self4
thf(fact_2266_nonzero__divide__mult__cancel__left,axiom,
    ! [A3: real,B3: real] :
      ( ( A3 != zero_zero_real )
     => ( ( divide_divide_real @ A3 @ ( times_times_real @ A3 @ B3 ) )
        = ( divide_divide_real @ one_one_real @ B3 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2267_nonzero__divide__mult__cancel__left,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3 != zero_zero_rat )
     => ( ( divide_divide_rat @ A3 @ ( times_times_rat @ A3 @ B3 ) )
        = ( divide_divide_rat @ one_one_rat @ B3 ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2268_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ B3 @ ( times_times_real @ A3 @ B3 ) )
        = ( divide_divide_real @ one_one_real @ A3 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2269_nonzero__divide__mult__cancel__right,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ B3 @ ( times_times_rat @ A3 @ B3 ) )
        = ( divide_divide_rat @ one_one_rat @ A3 ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2270_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_2271_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2272_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2273_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_2274_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_2275_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_2276_power__add__numeral,axiom,
    ! [A3: complex,M: num,N2: num] :
      ( ( times_times_complex @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2277_power__add__numeral,axiom,
    ! [A3: code_integer,M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2278_power__add__numeral,axiom,
    ! [A3: real,M: num,N2: num] :
      ( ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2279_power__add__numeral,axiom,
    ! [A3: rat,M: num,N2: num] :
      ( ( times_times_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2280_power__add__numeral,axiom,
    ! [A3: nat,M: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2281_power__add__numeral,axiom,
    ! [A3: int,M: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2282_power__add__numeral2,axiom,
    ! [A3: complex,M: num,N2: num,B3: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_times_complex @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2283_power__add__numeral2,axiom,
    ! [A3: code_integer,M: num,N2: num,B3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2284_power__add__numeral2,axiom,
    ! [A3: real,M: num,N2: num,B3: real] :
      ( ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2285_power__add__numeral2,axiom,
    ! [A3: rat,M: num,N2: num,B3: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_times_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2286_power__add__numeral2,axiom,
    ! [A3: nat,M: num,N2: num,B3: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_times_nat @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2287_power__add__numeral2,axiom,
    ! [A3: int,M: num,N2: num,B3: int] :
      ( ( times_times_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ N2 ) ) @ B3 ) )
      = ( times_times_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B3 ) ) ).

% power_add_numeral2
thf(fact_2288_mult__not__zero,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( times_7065122842183080059l_num1 @ A3 @ B3 )
       != zero_z3563351764282998399l_num1 )
     => ( ( A3 != zero_z3563351764282998399l_num1 )
        & ( B3 != zero_z3563351764282998399l_num1 ) ) ) ).

% mult_not_zero
thf(fact_2289_mult__not__zero,axiom,
    ! [A3: real,B3: real] :
      ( ( ( times_times_real @ A3 @ B3 )
       != zero_zero_real )
     => ( ( A3 != zero_zero_real )
        & ( B3 != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2290_mult__not__zero,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( times_times_rat @ A3 @ B3 )
       != zero_zero_rat )
     => ( ( A3 != zero_zero_rat )
        & ( B3 != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2291_mult__not__zero,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( times_times_nat @ A3 @ B3 )
       != zero_zero_nat )
     => ( ( A3 != zero_zero_nat )
        & ( B3 != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2292_mult__not__zero,axiom,
    ! [A3: int,B3: int] :
      ( ( ( times_times_int @ A3 @ B3 )
       != zero_zero_int )
     => ( ( A3 != zero_zero_int )
        & ( B3 != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2293_divisors__zero,axiom,
    ! [A3: real,B3: real] :
      ( ( ( times_times_real @ A3 @ B3 )
        = zero_zero_real )
     => ( ( A3 = zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2294_divisors__zero,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( times_times_rat @ A3 @ B3 )
        = zero_zero_rat )
     => ( ( A3 = zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2295_divisors__zero,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( times_times_nat @ A3 @ B3 )
        = zero_zero_nat )
     => ( ( A3 = zero_zero_nat )
        | ( B3 = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2296_divisors__zero,axiom,
    ! [A3: int,B3: int] :
      ( ( ( times_times_int @ A3 @ B3 )
        = zero_zero_int )
     => ( ( A3 = zero_zero_int )
        | ( B3 = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2297_no__zero__divisors,axiom,
    ! [A3: real,B3: real] :
      ( ( A3 != zero_zero_real )
     => ( ( B3 != zero_zero_real )
       => ( ( times_times_real @ A3 @ B3 )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2298_no__zero__divisors,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3 != zero_zero_rat )
     => ( ( B3 != zero_zero_rat )
       => ( ( times_times_rat @ A3 @ B3 )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2299_no__zero__divisors,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( B3 != zero_zero_nat )
       => ( ( times_times_nat @ A3 @ B3 )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2300_no__zero__divisors,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( B3 != zero_zero_int )
       => ( ( times_times_int @ A3 @ B3 )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2301_mult__left__cancel,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A3 )
          = ( times_times_real @ C @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_2302_mult__left__cancel,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A3 )
          = ( times_times_rat @ C @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_2303_mult__left__cancel,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A3 )
          = ( times_times_nat @ C @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_2304_mult__left__cancel,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A3 )
          = ( times_times_int @ C @ B3 ) )
        = ( A3 = B3 ) ) ) ).

% mult_left_cancel
thf(fact_2305_mult__right__cancel,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A3 @ C )
          = ( times_times_real @ B3 @ C ) )
        = ( A3 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_2306_mult__right__cancel,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A3 @ C )
          = ( times_times_rat @ B3 @ C ) )
        = ( A3 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_2307_mult__right__cancel,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A3 @ C )
          = ( times_times_nat @ B3 @ C ) )
        = ( A3 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_2308_mult__right__cancel,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A3 @ C )
          = ( times_times_int @ B3 @ C ) )
        = ( A3 = B3 ) ) ) ).

% mult_right_cancel
thf(fact_2309_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_2310_ring__class_Oring__distribs_I2_J,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2311_ring__class_Oring__distribs_I2_J,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2312_ring__class_Oring__distribs_I2_J,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2313_ring__class_Oring__distribs_I1_J,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2314_ring__class_Oring__distribs_I1_J,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2315_ring__class_Oring__distribs_I1_J,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2316_comm__semiring__class_Odistrib,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2317_comm__semiring__class_Odistrib,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2318_comm__semiring__class_Odistrib,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2319_comm__semiring__class_Odistrib,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2320_distrib__left,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).

% distrib_left
thf(fact_2321_distrib__left,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).

% distrib_left
thf(fact_2322_distrib__left,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).

% distrib_left
thf(fact_2323_distrib__left,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).

% distrib_left
thf(fact_2324_distrib__right,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).

% distrib_right
thf(fact_2325_distrib__right,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).

% distrib_right
thf(fact_2326_distrib__right,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ).

% distrib_right
thf(fact_2327_distrib__right,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).

% distrib_right
thf(fact_2328_combine__common__factor,axiom,
    ! [A3: real,E2: real,B3: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A3 @ B3 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2329_combine__common__factor,axiom,
    ! [A3: rat,E2: rat,B3: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A3 @ B3 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2330_combine__common__factor,axiom,
    ! [A3: nat,E2: nat,B3: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A3 @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B3 @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A3 @ B3 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2331_combine__common__factor,axiom,
    ! [A3: int,E2: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A3 @ B3 ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2332_comm__monoid__mult__class_Omult__1,axiom,
    ! [A3: uint32] :
      ( ( times_times_uint32 @ one_one_uint32 @ A3 )
      = A3 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2333_comm__monoid__mult__class_Omult__1,axiom,
    ! [A3: real] :
      ( ( times_times_real @ one_one_real @ A3 )
      = A3 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2334_comm__monoid__mult__class_Omult__1,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ one_one_rat @ A3 )
      = A3 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2335_comm__monoid__mult__class_Omult__1,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ one_one_nat @ A3 )
      = A3 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2336_comm__monoid__mult__class_Omult__1,axiom,
    ! [A3: int] :
      ( ( times_times_int @ one_one_int @ A3 )
      = A3 ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2337_mult_Ocomm__neutral,axiom,
    ! [A3: uint32] :
      ( ( times_times_uint32 @ A3 @ one_one_uint32 )
      = A3 ) ).

% mult.comm_neutral
thf(fact_2338_mult_Ocomm__neutral,axiom,
    ! [A3: real] :
      ( ( times_times_real @ A3 @ one_one_real )
      = A3 ) ).

% mult.comm_neutral
thf(fact_2339_mult_Ocomm__neutral,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ A3 @ one_one_rat )
      = A3 ) ).

% mult.comm_neutral
thf(fact_2340_mult_Ocomm__neutral,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ A3 @ one_one_nat )
      = A3 ) ).

% mult.comm_neutral
thf(fact_2341_mult_Ocomm__neutral,axiom,
    ! [A3: int] :
      ( ( times_times_int @ A3 @ one_one_int )
      = A3 ) ).

% mult.comm_neutral
thf(fact_2342_left__diff__distrib,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ).

% left_diff_distrib
thf(fact_2343_left__diff__distrib,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ).

% left_diff_distrib
thf(fact_2344_left__diff__distrib,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ).

% left_diff_distrib
thf(fact_2345_right__diff__distrib,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).

% right_diff_distrib
thf(fact_2346_right__diff__distrib,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).

% right_diff_distrib
thf(fact_2347_right__diff__distrib,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).

% right_diff_distrib
thf(fact_2348_left__diff__distrib_H,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( times_times_real @ ( minus_minus_real @ B3 @ C ) @ A3 )
      = ( minus_minus_real @ ( times_times_real @ B3 @ A3 ) @ ( times_times_real @ C @ A3 ) ) ) ).

% left_diff_distrib'
thf(fact_2349_left__diff__distrib_H,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B3 @ C ) @ A3 )
      = ( minus_minus_rat @ ( times_times_rat @ B3 @ A3 ) @ ( times_times_rat @ C @ A3 ) ) ) ).

% left_diff_distrib'
thf(fact_2350_left__diff__distrib_H,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B3 @ C ) @ A3 )
      = ( minus_minus_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C @ A3 ) ) ) ).

% left_diff_distrib'
thf(fact_2351_left__diff__distrib_H,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( times_times_int @ ( minus_minus_int @ B3 @ C ) @ A3 )
      = ( minus_minus_int @ ( times_times_int @ B3 @ A3 ) @ ( times_times_int @ C @ A3 ) ) ) ).

% left_diff_distrib'
thf(fact_2352_right__diff__distrib_H,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( times_times_real @ A3 @ ( minus_minus_real @ B3 @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A3 @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2353_right__diff__distrib_H,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( times_times_rat @ A3 @ ( minus_minus_rat @ B3 @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A3 @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2354_right__diff__distrib_H,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( times_times_nat @ A3 @ ( minus_minus_nat @ B3 @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2355_right__diff__distrib_H,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( times_times_int @ A3 @ ( minus_minus_int @ B3 @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) ) ) ).

% right_diff_distrib'
thf(fact_2356_times__divide__times__eq,axiom,
    ! [X2: real,Y2: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2357_times__divide__times__eq,axiom,
    ! [X2: rat,Y2: rat,Z: rat,W: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2358_divide__divide__times__eq,axiom,
    ! [X2: real,Y2: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X2 @ W ) @ ( times_times_real @ Y2 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2359_divide__divide__times__eq,axiom,
    ! [X2: rat,Y2: rat,Z: rat,W: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ Z @ W ) )
      = ( divide_divide_rat @ ( times_times_rat @ X2 @ W ) @ ( times_times_rat @ Y2 @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2360_divide__divide__eq__left_H,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A3 @ B3 ) @ C )
      = ( divide_divide_real @ A3 @ ( times_times_real @ C @ B3 ) ) ) ).

% divide_divide_eq_left'
thf(fact_2361_divide__divide__eq__left_H,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A3 @ B3 ) @ C )
      = ( divide_divide_rat @ A3 @ ( times_times_rat @ C @ B3 ) ) ) ).

% divide_divide_eq_left'
thf(fact_2362_power__commutes,axiom,
    ! [A3: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A3 @ N2 ) @ A3 )
      = ( times_times_complex @ A3 @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2363_power__commutes,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ A3 )
      = ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2364_power__commutes,axiom,
    ! [A3: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ A3 @ N2 ) @ A3 )
      = ( times_times_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2365_power__commutes,axiom,
    ! [A3: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ A3 @ N2 ) @ A3 )
      = ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2366_power__commutes,axiom,
    ! [A3: nat,N2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A3 @ N2 ) @ A3 )
      = ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2367_power__commutes,axiom,
    ! [A3: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ A3 @ N2 ) @ A3 )
      = ( times_times_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) ) ).

% power_commutes
thf(fact_2368_power__mult__distrib,axiom,
    ! [A3: complex,B3: complex,N2: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A3 @ B3 ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ A3 @ N2 ) @ ( power_power_complex @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2369_power__mult__distrib,axiom,
    ! [A3: code_integer,B3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2370_power__mult__distrib,axiom,
    ! [A3: real,B3: real,N2: nat] :
      ( ( power_power_real @ ( times_times_real @ A3 @ B3 ) @ N2 )
      = ( times_times_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2371_power__mult__distrib,axiom,
    ! [A3: rat,B3: rat,N2: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A3 @ B3 ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2372_power__mult__distrib,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A3 @ B3 ) @ N2 )
      = ( times_times_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2373_power__mult__distrib,axiom,
    ! [A3: int,B3: int,N2: nat] :
      ( ( power_power_int @ ( times_times_int @ A3 @ B3 ) @ N2 )
      = ( times_times_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2374_power__commuting__commutes,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( ( times_times_complex @ X2 @ Y2 )
        = ( times_times_complex @ Y2 @ X2 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X2 @ N2 ) @ Y2 )
        = ( times_times_complex @ Y2 @ ( power_power_complex @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2375_power__commuting__commutes,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat] :
      ( ( ( times_3573771949741848930nteger @ X2 @ Y2 )
        = ( times_3573771949741848930nteger @ Y2 @ X2 ) )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ Y2 )
        = ( times_3573771949741848930nteger @ Y2 @ ( power_8256067586552552935nteger @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2376_power__commuting__commutes,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( ( times_times_real @ X2 @ Y2 )
        = ( times_times_real @ Y2 @ X2 ) )
     => ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ Y2 )
        = ( times_times_real @ Y2 @ ( power_power_real @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2377_power__commuting__commutes,axiom,
    ! [X2: rat,Y2: rat,N2: nat] :
      ( ( ( times_times_rat @ X2 @ Y2 )
        = ( times_times_rat @ Y2 @ X2 ) )
     => ( ( times_times_rat @ ( power_power_rat @ X2 @ N2 ) @ Y2 )
        = ( times_times_rat @ Y2 @ ( power_power_rat @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2378_power__commuting__commutes,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = ( times_times_nat @ Y2 @ X2 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ Y2 )
        = ( times_times_nat @ Y2 @ ( power_power_nat @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2379_power__commuting__commutes,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( ( times_times_int @ X2 @ Y2 )
        = ( times_times_int @ Y2 @ X2 ) )
     => ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ Y2 )
        = ( times_times_int @ Y2 @ ( power_power_int @ X2 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2380_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_2381_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2382_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2383_power__mult,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A3 @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2384_power__mult,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( power_power_int @ A3 @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A3 @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2385_power__mult,axiom,
    ! [A3: real,M: nat,N2: nat] :
      ( ( power_power_real @ A3 @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A3 @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2386_power__mult,axiom,
    ! [A3: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A3 @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A3 @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2387_power__mult,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( times_times_nat @ M @ N2 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2388_plus__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( plus_plus_int @ zero_zero_int @ L2 )
      = L2 ) ).

% plus_int_code(2)
thf(fact_2389_plus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( plus_plus_int @ K @ zero_zero_int )
      = K ) ).

% plus_int_code(1)
thf(fact_2390_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_2391_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2392_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_2393_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2394_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2395_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_2396_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_2397_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_2398_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2399_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2400_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_2401_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_2402_div__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_2403_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_2404_lambda__zero,axiom,
    ( ( ^ [H: word_N3645301735248828278l_num1] : zero_z3563351764282998399l_num1 )
    = ( times_7065122842183080059l_num1 @ zero_z3563351764282998399l_num1 ) ) ).

% lambda_zero
thf(fact_2405_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_2406_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_2407_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_2408_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_2409_lambda__one,axiom,
    ( ( ^ [X: uint32] : X )
    = ( times_times_uint32 @ one_one_uint32 ) ) ).

% lambda_one
thf(fact_2410_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_2411_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_2412_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_2413_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_2414_mult__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2415_mult__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2416_mult__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2417_mult__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2418_mult__mono_H,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2419_mult__mono_H,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2420_mult__mono_H,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2421_mult__mono_H,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2422_zero__le__square,axiom,
    ! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).

% zero_le_square
thf(fact_2423_zero__le__square,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ A3 ) ) ).

% zero_le_square
thf(fact_2424_zero__le__square,axiom,
    ! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ A3 ) ) ).

% zero_le_square
thf(fact_2425_split__mult__pos__le,axiom,
    ! [A3: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_2426_split__mult__pos__le,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_2427_split__mult__pos__le,axiom,
    ! [A3: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_eq_int @ A3 @ zero_zero_int )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ).

% split_mult_pos_le
thf(fact_2428_mult__left__mono__neg,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2429_mult__left__mono__neg,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2430_mult__left__mono__neg,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2431_mult__nonpos__nonpos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2432_mult__nonpos__nonpos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2433_mult__nonpos__nonpos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2434_mult__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_2435_mult__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_2436_mult__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_2437_mult__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_left_mono
thf(fact_2438_mult__right__mono__neg,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2439_mult__right__mono__neg,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2440_mult__right__mono__neg,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2441_mult__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2442_mult__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2443_mult__right__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2444_mult__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2445_mult__le__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2446_mult__le__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2447_mult__le__0__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A3 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).

% mult_le_0_iff
thf(fact_2448_split__mult__neg__le,axiom,
    ! [A3: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2449_split__mult__neg__le,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2450_split__mult__neg__le,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
          & ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2451_split__mult__neg__le,axiom,
    ! [A3: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A3 @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2452_mult__nonneg__nonneg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2453_mult__nonneg__nonneg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2454_mult__nonneg__nonneg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2455_mult__nonneg__nonneg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2456_mult__nonneg__nonpos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2457_mult__nonneg__nonpos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2458_mult__nonneg__nonpos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2459_mult__nonneg__nonpos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2460_mult__nonpos__nonneg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2461_mult__nonpos__nonneg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2462_mult__nonpos__nonneg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2463_mult__nonpos__nonneg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2464_mult__nonneg__nonpos2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2465_mult__nonneg__nonpos2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B3 @ A3 ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2466_mult__nonneg__nonpos2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2467_mult__nonneg__nonpos2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2468_zero__le__mult__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_eq_real @ A3 @ zero_zero_real )
          & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2469_zero__le__mult__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
          & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2470_zero__le__mult__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
          & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_eq_int @ A3 @ zero_zero_int )
          & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2471_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2472_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2473_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2474_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2475_mult__neg__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_2476_mult__neg__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_2477_mult__neg__neg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% mult_neg_neg
thf(fact_2478_not__square__less__zero,axiom,
    ! [A3: real] :
      ~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2479_not__square__less__zero,axiom,
    ! [A3: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A3 @ A3 ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2480_not__square__less__zero,axiom,
    ! [A3: int] :
      ~ ( ord_less_int @ ( times_times_int @ A3 @ A3 ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2481_mult__less__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ B3 @ zero_zero_real ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2482_mult__less__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2483_mult__less__0__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A3 )
          & ( ord_less_int @ B3 @ zero_zero_int ) )
        | ( ( ord_less_int @ A3 @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).

% mult_less_0_iff
thf(fact_2484_mult__neg__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2485_mult__neg__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2486_mult__neg__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2487_mult__neg__pos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2488_mult__pos__neg,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2489_mult__pos__neg,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2490_mult__pos__neg,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2491_mult__pos__neg,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2492_mult__pos__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_2493_mult__pos__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ zero_zero_rat @ B3 )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_2494_mult__pos__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_2495_mult__pos__pos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% mult_pos_pos
thf(fact_2496_mult__pos__neg2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ B3 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2497_mult__pos__neg2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ B3 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B3 @ A3 ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2498_mult__pos__neg2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ B3 @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2499_mult__pos__neg2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B3 @ A3 ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2500_zero__less__mult__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A3 )
          & ( ord_less_real @ zero_zero_real @ B3 ) )
        | ( ( ord_less_real @ A3 @ zero_zero_real )
          & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2501_zero__less__mult__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A3 )
          & ( ord_less_rat @ zero_zero_rat @ B3 ) )
        | ( ( ord_less_rat @ A3 @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2502_zero__less__mult__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A3 )
          & ( ord_less_int @ zero_zero_int @ B3 ) )
        | ( ( ord_less_int @ A3 @ zero_zero_int )
          & ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2503_zero__less__mult__pos,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_2504_zero__less__mult__pos,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A3 )
       => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_2505_zero__less__mult__pos,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A3 )
       => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_2506_zero__less__mult__pos,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A3 @ B3 ) )
     => ( ( ord_less_int @ zero_zero_int @ A3 )
       => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).

% zero_less_mult_pos
thf(fact_2507_zero__less__mult__pos2,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2508_zero__less__mult__pos2,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B3 @ A3 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A3 )
       => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2509_zero__less__mult__pos2,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A3 )
       => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2510_zero__less__mult__pos2,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A3 ) )
     => ( ( ord_less_int @ zero_zero_int @ A3 )
       => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).

% zero_less_mult_pos2
thf(fact_2511_mult__less__cancel__left__neg,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_real @ B3 @ A3 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2512_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_rat @ B3 @ A3 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2513_mult__less__cancel__left__neg,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_int @ B3 @ A3 ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2514_mult__less__cancel__left__pos,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_real @ A3 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2515_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_rat @ A3 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2516_mult__less__cancel__left__pos,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_int @ A3 @ B3 ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2517_mult__strict__left__mono__neg,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2518_mult__strict__left__mono__neg,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2519_mult__strict__left__mono__neg,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_int @ B3 @ A3 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2520_mult__strict__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2521_mult__strict__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2522_mult__strict__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2523_mult__strict__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2524_mult__less__cancel__left__disj,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A3 @ B3 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2525_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A3 @ B3 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2526_mult__less__cancel__left__disj,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A3 @ B3 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2527_mult__strict__right__mono__neg,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2528_mult__strict__right__mono__neg,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2529_mult__strict__right__mono__neg,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_int @ B3 @ A3 )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2530_mult__strict__right__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2531_mult__strict__right__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2532_mult__strict__right__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2533_mult__strict__right__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2534_mult__less__cancel__right__disj,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A3 @ B3 ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2535_mult__less__cancel__right__disj,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A3 @ B3 ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2536_mult__less__cancel__right__disj,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A3 @ B3 ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2537_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2538_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2539_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2540_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2541_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2542_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2543_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2544_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2545_mult__numeral__1,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ one ) @ A3 )
      = A3 ) ).

% mult_numeral_1
thf(fact_2546_mult__numeral__1,axiom,
    ! [A3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A3 )
      = A3 ) ).

% mult_numeral_1
thf(fact_2547_mult__numeral__1,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A3 )
      = A3 ) ).

% mult_numeral_1
thf(fact_2548_mult__numeral__1,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A3 )
      = A3 ) ).

% mult_numeral_1
thf(fact_2549_mult__numeral__1,axiom,
    ! [A3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A3 )
      = A3 ) ).

% mult_numeral_1
thf(fact_2550_mult__numeral__1__right,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ one ) )
      = A3 ) ).

% mult_numeral_1_right
thf(fact_2551_mult__numeral__1__right,axiom,
    ! [A3: real] :
      ( ( times_times_real @ A3 @ ( numeral_numeral_real @ one ) )
      = A3 ) ).

% mult_numeral_1_right
thf(fact_2552_mult__numeral__1__right,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ A3 @ ( numeral_numeral_rat @ one ) )
      = A3 ) ).

% mult_numeral_1_right
thf(fact_2553_mult__numeral__1__right,axiom,
    ! [A3: nat] :
      ( ( times_times_nat @ A3 @ ( numeral_numeral_nat @ one ) )
      = A3 ) ).

% mult_numeral_1_right
thf(fact_2554_mult__numeral__1__right,axiom,
    ! [A3: int] :
      ( ( times_times_int @ A3 @ ( numeral_numeral_int @ one ) )
      = A3 ) ).

% mult_numeral_1_right
thf(fact_2555_frac__eq__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X2 @ Y2 )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X2 @ Z )
            = ( times_times_real @ W @ Y2 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2556_frac__eq__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X2 @ Y2 )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X2 @ Z )
            = ( times_times_rat @ W @ Y2 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_2557_divide__eq__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ( divide_divide_real @ B3 @ C )
        = A3 )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( times_times_real @ A3 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_2558_divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = A3 )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A3 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_2559_eq__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( A3
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A3 @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_2560_eq__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( A3
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A3 @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_2561_divide__eq__imp,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( C != zero_zero_real )
     => ( ( B3
          = ( times_times_real @ A3 @ C ) )
       => ( ( divide_divide_real @ B3 @ C )
          = A3 ) ) ) ).

% divide_eq_imp
thf(fact_2562_divide__eq__imp,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B3
          = ( times_times_rat @ A3 @ C ) )
       => ( ( divide_divide_rat @ B3 @ C )
          = A3 ) ) ) ).

% divide_eq_imp
thf(fact_2563_eq__divide__imp,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A3 @ C )
          = B3 )
       => ( A3
          = ( divide_divide_real @ B3 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2564_eq__divide__imp,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A3 @ C )
          = B3 )
       => ( A3
          = ( divide_divide_rat @ B3 @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_2565_nonzero__divide__eq__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B3 @ C )
          = A3 )
        = ( B3
          = ( times_times_real @ A3 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2566_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B3 @ C )
          = A3 )
        = ( B3
          = ( times_times_rat @ A3 @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_2567_nonzero__eq__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( C != zero_zero_real )
     => ( ( A3
          = ( divide_divide_real @ B3 @ C ) )
        = ( ( times_times_real @ A3 @ C )
          = B3 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2568_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A3
          = ( divide_divide_rat @ B3 @ C ) )
        = ( ( times_times_rat @ A3 @ C )
          = B3 ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_2569_eq__add__iff1,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2570_eq__add__iff1,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2571_eq__add__iff1,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2572_eq__add__iff2,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2573_eq__add__iff2,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2574_eq__add__iff2,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2575_square__diff__square__factored,axiom,
    ! [X2: real,Y2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) )
      = ( times_times_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_real @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_2576_square__diff__square__factored,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( minus_minus_rat @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_2577_square__diff__square__factored,axiom,
    ! [X2: int,Y2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) )
      = ( times_times_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( minus_minus_int @ X2 @ Y2 ) ) ) ).

% square_diff_square_factored
thf(fact_2578_left__right__inverse__power,axiom,
    ! [X2: uint32,Y2: uint32,N2: nat] :
      ( ( ( times_times_uint32 @ X2 @ Y2 )
        = one_one_uint32 )
     => ( ( times_times_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ ( power_power_uint32 @ Y2 @ N2 ) )
        = one_one_uint32 ) ) ).

% left_right_inverse_power
thf(fact_2579_left__right__inverse__power,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( ( times_times_complex @ X2 @ Y2 )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_2580_left__right__inverse__power,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat] :
      ( ( ( times_3573771949741848930nteger @ X2 @ Y2 )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ ( power_8256067586552552935nteger @ Y2 @ N2 ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_2581_left__right__inverse__power,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( ( times_times_real @ X2 @ Y2 )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_2582_left__right__inverse__power,axiom,
    ! [X2: rat,Y2: rat,N2: nat] :
      ( ( ( times_times_rat @ X2 @ Y2 )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X2 @ N2 ) @ ( power_power_rat @ Y2 @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2583_left__right__inverse__power,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( ( times_times_nat @ X2 @ Y2 )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2584_left__right__inverse__power,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( ( times_times_int @ X2 @ Y2 )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2585_power__Suc2,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ A3 @ ( suc @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2586_power__Suc2,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2587_power__Suc2,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ A3 @ ( suc @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2588_power__Suc2,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ A3 @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2589_power__Suc2,axiom,
    ! [A3: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( suc @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2590_power__Suc2,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ A3 @ ( suc @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A3 @ N2 ) @ A3 ) ) ).

% power_Suc2
thf(fact_2591_power__Suc,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ A3 @ ( suc @ N2 ) )
      = ( times_times_complex @ A3 @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2592_power__Suc,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( suc @ N2 ) )
      = ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2593_power__Suc,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ A3 @ ( suc @ N2 ) )
      = ( times_times_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2594_power__Suc,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ A3 @ ( suc @ N2 ) )
      = ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2595_power__Suc,axiom,
    ! [A3: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( suc @ N2 ) )
      = ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2596_power__Suc,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ A3 @ ( suc @ N2 ) )
      = ( times_times_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) ) ).

% power_Suc
thf(fact_2597_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_2598_power__add,axiom,
    ! [A3: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A3 @ M ) @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2599_power__add,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2600_power__add,axiom,
    ! [A3: real,M: nat,N2: nat] :
      ( ( power_power_real @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2601_power__add,axiom,
    ! [A3: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A3 @ M ) @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2602_power__add,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2603_power__add,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( power_power_int @ A3 @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) ) ) ).

% power_add
thf(fact_2604_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_2605_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_2606_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_2607_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_2608_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_2609_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_2610_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_2611_mlex__snd__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,B4: nat,N3: nat] :
      ( ( A3 = A5 )
     => ( ( ord_less_nat @ B3 @ B4 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A3 @ N3 ) @ B3 ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B4 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_2612_mlex__fst__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,N3: nat,B4: nat] :
      ( ( ord_less_nat @ A3 @ A5 )
     => ( ( ord_less_nat @ B3 @ N3 )
       => ( ( ord_less_nat @ B4 @ N3 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A3 @ N3 ) @ B3 ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B4 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_2613_mlex__bound,axiom,
    ! [A3: nat,A4: nat,B3: nat,N3: nat] :
      ( ( ord_less_nat @ A3 @ A4 )
     => ( ( ord_less_nat @ B3 @ N3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A3 @ N3 ) @ B3 ) @ ( times_times_nat @ A4 @ N3 ) ) ) ) ).

% mlex_bound
thf(fact_2614_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_2615_mlex__leI,axiom,
    ! [A3: nat,A5: nat,B3: nat,B4: nat,N3: nat] :
      ( ( ord_less_eq_nat @ A3 @ A5 )
     => ( ( ord_less_eq_nat @ B3 @ B4 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A3 @ N3 ) @ B3 ) @ ( plus_plus_nat @ ( times_times_nat @ A5 @ N3 ) @ B4 ) ) ) ) ).

% mlex_leI
thf(fact_2616_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_2617_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_2618_div__mult__le,axiom,
    ! [A3: nat,B3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ A3 ) ).

% div_mult_le
thf(fact_2619_odd__nonzero,axiom,
    ! [Z: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_2620_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_2621_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I2: int] :
              ( ( ord_less_int @ K @ I2 )
             => ( ( P @ I2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_2622_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_2623_power__odd__eq,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_complex @ A3 @ ( power_power_complex @ ( power_power_complex @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2624_power__odd__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2625_power__odd__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_real @ A3 @ ( power_power_real @ ( power_power_real @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2626_power__odd__eq,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A3 @ ( power_power_rat @ ( power_power_rat @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2627_power__odd__eq,axiom,
    ! [A3: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_nat @ A3 @ ( power_power_nat @ ( power_power_nat @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2628_power__odd__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_int @ A3 @ ( power_power_int @ ( power_power_int @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_2629_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y2: extended_enat,X2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y2 )
     => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y2 @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y2 ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_2630_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_2631_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_2632_mult__le__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A3 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2633_mult__le__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A3 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2634_mult__le__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A3 @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_left
thf(fact_2635_mult__le__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A3 @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2636_mult__le__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A3 @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2637_mult__le__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A3 @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% mult_le_cancel_right
thf(fact_2638_mult__left__less__imp__less,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A3 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2639_mult__left__less__imp__less,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A3 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2640_mult__left__less__imp__less,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A3 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2641_mult__left__less__imp__less,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A3 @ B3 ) ) ) ).

% mult_left_less_imp_less
thf(fact_2642_mult__strict__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2643_mult__strict__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2644_mult__strict__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2645_mult__strict__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_2646_mult__less__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A3 @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2647_mult__less__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A3 @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2648_mult__less__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A3 @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_left
thf(fact_2649_mult__right__less__imp__less,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A3 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2650_mult__right__less__imp__less,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A3 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2651_mult__right__less__imp__less,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A3 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2652_mult__right__less__imp__less,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A3 @ B3 ) ) ) ).

% mult_right_less_imp_less
thf(fact_2653_mult__strict__mono_H,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2654_mult__strict__mono_H,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2655_mult__strict__mono_H,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2656_mult__strict__mono_H,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_2657_mult__less__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A3 @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2658_mult__less__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A3 @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2659_mult__less__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A3 @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ A3 ) ) ) ) ).

% mult_less_cancel_right
thf(fact_2660_mult__le__cancel__left__neg,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_eq_real @ B3 @ A3 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2661_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_eq_rat @ B3 @ A3 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2662_mult__le__cancel__left__neg,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_eq_int @ B3 @ A3 ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_2663_mult__le__cancel__left__pos,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
        = ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2664_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
        = ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2665_mult__le__cancel__left__pos,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
        = ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_2666_mult__left__le__imp__le,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2667_mult__left__le__imp__le,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2668_mult__left__le__imp__le,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2669_mult__left__le__imp__le,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% mult_left_le_imp_le
thf(fact_2670_mult__right__le__imp__le,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2671_mult__right__le__imp__le,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2672_mult__right__le__imp__le,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2673_mult__right__le__imp__le,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% mult_right_le_imp_le
thf(fact_2674_mult__le__less__imp__less,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2675_mult__le__less__imp__less,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A3 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2676_mult__le__less__imp__less,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A3 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2677_mult__le__less__imp__less,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_2678_mult__less__le__imp__less,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2679_mult__less__le__imp__less,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2680_mult__less__le__imp__less,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2681_mult__less__le__imp__less,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_2682_sum__squares__le__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2683_sum__squares__le__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat )
      = ( ( X2 = zero_zero_rat )
        & ( Y2 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2684_sum__squares__le__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int )
      = ( ( X2 = zero_zero_int )
        & ( Y2 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_2685_sum__squares__ge__zero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2686_sum__squares__ge__zero,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2687_sum__squares__ge__zero,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) ) ).

% sum_squares_ge_zero
thf(fact_2688_mult__left__le__one__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2689_mult__left__le__one__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2690_mult__left__le__one__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ord_less_eq_int @ Y2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y2 @ X2 ) @ X2 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_2691_mult__right__le__one__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2692_mult__right__le__one__le,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
       => ( ( ord_less_eq_rat @ Y2 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2693_mult__right__le__one__le,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( ord_less_eq_int @ Y2 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y2 ) @ X2 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_2694_mult__le__one,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ( ord_less_eq_real @ B3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_2695_mult__le__one,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ( ord_less_eq_rat @ B3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ B3 ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_2696_mult__le__one,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_2697_mult__le__one,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_2698_mult__left__le,axiom,
    ! [C: real,A3: real] :
      ( ( ord_less_eq_real @ C @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ A3 ) ) ) ).

% mult_left_le
thf(fact_2699_mult__left__le,axiom,
    ! [C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ A3 ) ) ) ).

% mult_left_le
thf(fact_2700_mult__left__le,axiom,
    ! [C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ C @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).

% mult_left_le
thf(fact_2701_mult__left__le,axiom,
    ! [C: int,A3: int] :
      ( ( ord_less_eq_int @ C @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A3 )
       => ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ A3 ) ) ) ).

% mult_left_le
thf(fact_2702_sum__squares__gt__zero__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_real )
        | ( Y2 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2703_sum__squares__gt__zero__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_rat )
        | ( Y2 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2704_sum__squares__gt__zero__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) )
      = ( ( X2 != zero_zero_int )
        | ( Y2 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_2705_not__sum__squares__lt__zero,axiom,
    ! [X2: real,Y2: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y2 @ Y2 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_2706_not__sum__squares__lt__zero,axiom,
    ! [X2: rat,Y2: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y2 @ Y2 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_2707_not__sum__squares__lt__zero,axiom,
    ! [X2: int,Y2: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y2 @ Y2 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_2708_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2709_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_2710_divide__less__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2711_divide__less__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_2712_less__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2713_less__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_2714_neg__divide__less__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
        = ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).

% neg_divide_less_eq
thf(fact_2715_neg__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
        = ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).

% neg_divide_less_eq
thf(fact_2716_neg__less__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2717_neg__less__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_2718_pos__divide__less__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
        = ( ord_less_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2719_pos__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
        = ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_2720_pos__less__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).

% pos_less_divide_eq
thf(fact_2721_pos__less__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).

% pos_less_divide_eq
thf(fact_2722_mult__imp__div__pos__less,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
       => ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2723_mult__imp__div__pos__less,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_2724_mult__imp__less__div__pos,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2725_mult__imp__less__div__pos,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y2 ) @ X2 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_2726_divide__strict__left__mono,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2727_divide__strict__left__mono,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_2728_divide__strict__left__mono__neg,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2729_divide__strict__left__mono__neg,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_2730_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B3 @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( 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_2731_divide__eq__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( 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_2732_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2733_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_2734_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2735_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2736_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2737_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2738_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2739_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2740_less__add__iff1,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2741_less__add__iff1,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2742_less__add__iff1,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A3 @ B3 ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2743_less__add__iff2,axiom,
    ! [A3: real,E2: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A3 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2744_less__add__iff2,axiom,
    ! [A3: rat,E2: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2745_less__add__iff2,axiom,
    ! [A3: int,E2: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A3 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A3 ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2746_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
          = ( divide_divide_real @ ( plus_plus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2747_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
          = ( divide_divide_rat @ ( plus_plus_rat @ A3 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_2748_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
          = A3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2749_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
          = A3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_2750_add__frac__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2751_add__frac__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_2752_add__frac__num,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_frac_num
thf(fact_2753_add__frac__num,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_frac_num
thf(fact_2754_add__num__frac,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_num_frac
thf(fact_2755_add__num__frac,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) ) @ Y2 ) ) ) ).

% add_num_frac
thf(fact_2756_add__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y2 @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2757_add__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y2 @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_2758_divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y2 )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2759_divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_2760_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
          = A3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A3 @ ( divide_divide_real @ B3 @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2761_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
          = A3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A3 @ ( divide_divide_rat @ B3 @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A3 @ Z ) @ B3 ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_2762_diff__frac__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2763_diff__frac__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_2764_diff__divide__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y2 @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2765_diff__divide__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y2 @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y2 ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_2766_divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y2 )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2767_divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y2 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_2768_square__diff__one__factored,axiom,
    ! [X2: uint32] :
      ( ( minus_minus_uint32 @ ( times_times_uint32 @ X2 @ X2 ) @ one_one_uint32 )
      = ( times_times_uint32 @ ( plus_plus_uint32 @ X2 @ one_one_uint32 ) @ ( minus_minus_uint32 @ X2 @ one_one_uint32 ) ) ) ).

% square_diff_one_factored
thf(fact_2769_square__diff__one__factored,axiom,
    ! [X2: real] :
      ( ( minus_minus_real @ ( times_times_real @ X2 @ X2 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X2 @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_2770_square__diff__one__factored,axiom,
    ! [X2: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X2 @ X2 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_2771_square__diff__one__factored,axiom,
    ! [X2: int] :
      ( ( minus_minus_int @ ( times_times_int @ X2 @ X2 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X2 @ one_one_int ) @ ( minus_minus_int @ X2 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_2772_power__gt1__lemma,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2773_power__gt1__lemma,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ord_less_real @ one_one_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2774_power__gt1__lemma,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2775_power__gt1__lemma,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2776_power__gt1__lemma,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_2777_power__less__power__Suc,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A3 )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2778_power__less__power__Suc,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ord_less_real @ ( power_power_real @ A3 @ N2 ) @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2779_power__less__power__Suc,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A3 )
     => ( ord_less_rat @ ( power_power_rat @ A3 @ N2 ) @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2780_power__less__power__Suc,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A3 )
     => ( ord_less_nat @ ( power_power_nat @ A3 @ N2 ) @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2781_power__less__power__Suc,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ord_less_int @ ( power_power_int @ A3 @ N2 ) @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_2782_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_2783_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_2784_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_2785_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_2786_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_2787_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_2788_td__gal__lt,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
        = ( ord_less_nat @ ( divide_divide_nat @ A3 @ C ) @ B3 ) ) ) ).

% td_gal_lt
thf(fact_2789_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_2790_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_2791_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_2792_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_2793_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_2794_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_2795_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_2796_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_2797_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_2798_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I2: int] :
            ( ( ord_less_eq_int @ K @ I2 )
           => ( ( P @ I2 )
             => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
       => ( ! [I2: int] :
              ( ( ord_less_eq_int @ I2 @ K )
             => ( ( P @ I2 )
               => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_2799_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_2800_power__numeral__even,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2801_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_2802_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_2803_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_2804_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_2805_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
thf(fact_2806_field__le__mult__one__interval,axiom,
    ! [X2: real,Y2: real] :
      ( ! [Z4: real] :
          ( ( ord_less_real @ zero_zero_real @ Z4 )
         => ( ( ord_less_real @ Z4 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_real @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2807_field__le__mult__one__interval,axiom,
    ! [X2: rat,Y2: rat] :
      ( ! [Z4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z4 )
         => ( ( ord_less_rat @ Z4 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X2 ) @ Y2 ) ) )
     => ( ord_less_eq_rat @ X2 @ Y2 ) ) ).

% field_le_mult_one_interval
thf(fact_2808_mult__le__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2809_mult__le__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2810_mult__le__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_2811_mult__le__cancel__left2,axiom,
    ! [C: real,A3: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A3 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2812_mult__le__cancel__left2,axiom,
    ! [C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A3 ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A3 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A3 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2813_mult__le__cancel__left2,axiom,
    ! [C: int,A3: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A3 ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A3 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_2814_mult__le__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B3 ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2815_mult__le__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2816_mult__le__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B3 ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_2817_mult__le__cancel__right2,axiom,
    ! [A3: real,C: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ C )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A3 @ one_one_real ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2818_mult__le__cancel__right2,axiom,
    ! [A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A3 @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A3 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2819_mult__le__cancel__right2,axiom,
    ! [A3: int,C: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A3 @ C ) @ C )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A3 @ one_one_int ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A3 ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_2820_mult__less__cancel__left1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2821_mult__less__cancel__left1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B3 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2822_mult__less__cancel__left1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B3 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_2823_mult__less__cancel__left2,axiom,
    ! [C: real,A3: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A3 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2824_mult__less__cancel__left2,axiom,
    ! [C: rat,A3: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A3 ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A3 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A3 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2825_mult__less__cancel__left2,axiom,
    ! [C: int,A3: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A3 ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A3 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_2826_mult__less__cancel__right1,axiom,
    ! [C: real,B3: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B3 ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2827_mult__less__cancel__right1,axiom,
    ! [C: rat,B3: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B3 @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B3 ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2828_mult__less__cancel__right1,axiom,
    ! [C: int,B3: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B3 @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B3 ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_2829_mult__less__cancel__right2,axiom,
    ! [A3: real,C: real] :
      ( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ C )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A3 @ one_one_real ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2830_mult__less__cancel__right2,axiom,
    ! [A3: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A3 @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A3 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2831_mult__less__cancel__right2,axiom,
    ! [A3: int,C: int] :
      ( ( ord_less_int @ ( times_times_int @ A3 @ C ) @ C )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A3 @ one_one_int ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A3 ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_2832_convex__bound__le,axiom,
    ! [X2: real,A3: real,Y2: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X2 @ A3 )
     => ( ( ord_less_eq_real @ Y2 @ A3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2833_convex__bound__le,axiom,
    ! [X2: rat,A3: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X2 @ A3 )
     => ( ( ord_less_eq_rat @ Y2 @ A3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2834_convex__bound__le,axiom,
    ! [X2: int,A3: int,Y2: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X2 @ A3 )
     => ( ( ord_less_eq_int @ Y2 @ A3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_2835_divide__le__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2836_divide__le__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_2837_le__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2838_le__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_2839_divide__left__mono,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( ord_less_eq_real @ B3 @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2840_divide__left__mono,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_left_mono
thf(fact_2841_neg__divide__le__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
        = ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).

% neg_divide_le_eq
thf(fact_2842_neg__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).

% neg_divide_le_eq
thf(fact_2843_neg__le__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2844_neg__le__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_2845_pos__divide__le__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A3 )
        = ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2846_pos__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A3 )
        = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_2847_pos__le__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ B3 ) ) ) ).

% pos_le_divide_eq
thf(fact_2848_pos__le__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ B3 ) ) ) ).

% pos_le_divide_eq
thf(fact_2849_mult__imp__div__pos__le,axiom,
    ! [Y2: real,X2: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y2 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2850_mult__imp__div__pos__le,axiom,
    ! [Y2: rat,X2: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y2 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_2851_mult__imp__le__div__pos,axiom,
    ! [Y2: real,Z: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y2 ) @ X2 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2852_mult__imp__le__div__pos,axiom,
    ! [Y2: rat,Z: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y2 ) @ X2 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_2853_divide__left__mono__neg,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A3 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2854_divide__left__mono__neg,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A3 @ B3 ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A3 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_2855_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2856_divide__less__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_2857_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( 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_2858_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( 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_2859_mult__2,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ Z )
      = ( plus_p361126936061061375l_num1 @ Z @ Z ) ) ).

% mult_2
thf(fact_2860_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_2861_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_2862_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_2863_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_2864_mult__2__right,axiom,
    ! [Z: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ Z @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
      = ( plus_p361126936061061375l_num1 @ Z @ Z ) ) ).

% mult_2_right
thf(fact_2865_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_2866_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_2867_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_2868_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_2869_left__add__twice,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) )
      = ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).

% left_add_twice
thf(fact_2870_left__add__twice,axiom,
    ! [A3: real,B3: real] :
      ( ( plus_plus_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).

% left_add_twice
thf(fact_2871_left__add__twice,axiom,
    ! [A3: rat,B3: rat] :
      ( ( plus_plus_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).

% left_add_twice
thf(fact_2872_left__add__twice,axiom,
    ! [A3: nat,B3: nat] :
      ( ( plus_plus_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).

% left_add_twice
thf(fact_2873_left__add__twice,axiom,
    ! [A3: int,B3: int] :
      ( ( plus_plus_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) @ B3 ) ) ).

% left_add_twice
thf(fact_2874_frac__le__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_2875_frac__le__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_2876_power__Suc__less,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le6747313008572928689nteger @ A3 @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2877_power__Suc__less,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_real @ A3 @ one_one_real )
       => ( ord_less_real @ ( times_times_real @ A3 @ ( power_power_real @ A3 @ N2 ) ) @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2878_power__Suc__less,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_rat @ A3 @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A3 @ ( power_power_rat @ A3 @ N2 ) ) @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2879_power__Suc__less,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ A3 @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ A3 @ N2 ) ) @ ( power_power_nat @ A3 @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2880_power__Suc__less,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ A3 @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A3 @ ( power_power_int @ A3 @ N2 ) ) @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_2881_frac__less__eq,axiom,
    ! [Y2: real,Z: real,X2: real,W: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y2 ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W @ Y2 ) ) @ ( times_times_real @ Y2 @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_2882_frac__less__eq,axiom,
    ! [Y2: rat,Z: rat,X2: rat,W: rat] :
      ( ( Y2 != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y2 ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W @ Y2 ) ) @ ( times_times_rat @ Y2 @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_2883_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_2884_power4__eq__xxxx,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2885_power4__eq__xxxx,axiom,
    ! [X2: code_integer] :
      ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2886_power4__eq__xxxx,axiom,
    ! [X2: real] :
      ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2887_power4__eq__xxxx,axiom,
    ! [X2: rat] :
      ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2888_power4__eq__xxxx,axiom,
    ! [X2: nat] :
      ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2889_power4__eq__xxxx,axiom,
    ! [X2: int] :
      ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X2 @ X2 ) @ X2 ) @ X2 ) ) ).

% power4_eq_xxxx
thf(fact_2890_power2__eq__square,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2891_power2__eq__square,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2892_power2__eq__square,axiom,
    ! [A3: real] :
      ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2893_power2__eq__square,axiom,
    ! [A3: rat] :
      ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_rat @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2894_power2__eq__square,axiom,
    ! [A3: nat] :
      ( ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2895_power2__eq__square,axiom,
    ! [A3: int] :
      ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A3 @ A3 ) ) ).

% power2_eq_square
thf(fact_2896_power__even__eq,axiom,
    ! [A3: nat,N2: nat] :
      ( ( power_power_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_2897_power__even__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_2898_power__even__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_2899_power__even__eq,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_2900_power__even__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_2901_div__nat__eqI,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_2902_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_2903_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_2904_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_2905_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_2906_td__gal,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ C ) @ A3 )
        = ( ord_less_eq_nat @ B3 @ ( divide_divide_nat @ A3 @ C ) ) ) ) ).

% td_gal
thf(fact_2907_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_2908_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_2909_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_2910_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_2911_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,Va2: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
thf(fact_2912_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv3: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ Uu2 @ Uv3 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
thf(fact_2913_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv3: $o,Uw3: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uv3 @ Uw3 ) @ ( suc @ N2 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
thf(fact_2914_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
thf(fact_2915_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B3: $o] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
thf(fact_2916_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
thf(fact_2917_convex__bound__lt,axiom,
    ! [X2: real,A3: real,Y2: real,U: real,V: real] :
      ( ( ord_less_real @ X2 @ A3 )
     => ( ( ord_less_real @ Y2 @ A3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2918_convex__bound__lt,axiom,
    ! [X2: rat,A3: rat,Y2: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X2 @ A3 )
     => ( ( ord_less_rat @ Y2 @ A3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2919_convex__bound__lt,axiom,
    ! [X2: int,A3: int,Y2: int,U: int,V: int] :
      ( ( ord_less_int @ X2 @ A3 )
     => ( ( ord_less_int @ Y2 @ A3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y2 ) ) @ A3 ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_2920_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_2921_divide__le__eq__numeral_I1_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_2922_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( 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_2923_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( 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_2924_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_2925_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_2926_power__eq__if,axiom,
    ( power_power_uint32
    = ( ^ [P5: uint32,M3: nat] : ( if_uint32 @ ( M3 = zero_zero_nat ) @ one_one_uint32 @ ( times_times_uint32 @ P5 @ ( power_power_uint32 @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2927_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2928_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P5: code_integer,M3: nat] : ( if_Code_integer @ ( M3 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2929_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2930_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2931_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2932_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2933_power__minus__mult,axiom,
    ! [N2: nat,A3: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( power_power_complex @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2934_power__minus__mult,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2935_power__minus__mult,axiom,
    ! [N2: nat,A3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( power_power_real @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_power_real @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2936_power__minus__mult,axiom,
    ! [N2: nat,A3: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2937_power__minus__mult,axiom,
    ! [N2: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_nat @ ( power_power_nat @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_power_nat @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2938_power__minus__mult,axiom,
    ! [N2: nat,A3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( power_power_int @ A3 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A3 )
        = ( power_power_int @ A3 @ N2 ) ) ) ).

% power_minus_mult
thf(fact_2939_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_2940_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
thf(fact_2941_power2__sum,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2942_power2__sum,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( power_8256067586552552935nteger @ ( plus_p5714425477246183910nteger @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2943_power2__sum,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_2184487114949457152l_num1 @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2944_power2__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2945_power2__sum,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2946_power2__sum,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2947_power2__sum,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( plus_plus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_sum
thf(fact_2948_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A3: $o,Uw3: $o] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Leaf @ A3 @ Uw3 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
thf(fact_2949_zero__le__even__power_H,axiom,
    ! [A3: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2950_zero__le__even__power_H,axiom,
    ! [A3: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2951_zero__le__even__power_H,axiom,
    ! [A3: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2952_zero__le__even__power_H,axiom,
    ! [A3: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_2953_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
thf(fact_2954_nat__bit__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( P @ N4 )
           => ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
       => ( ! [N4: nat] :
              ( ( P @ N4 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_bit_induct
thf(fact_2955_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
thf(fact_2956_axxdiv2,axiom,
    ! [X2: int] :
      ( ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X2 ) @ X2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X2 )
      & ( ( divide_divide_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X2 ) @ X2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = X2 ) ) ).

% axxdiv2
thf(fact_2957_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_2958_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E: real] :
            ( ( ord_less_real @ zero_zero_real @ E )
            & ( ord_less_real @ E @ D1 )
            & ( ord_less_real @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2959_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2960_two__realpow__ge__one,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% two_realpow_ge_one
thf(fact_2961_power2__diff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2962_power2__diff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( power_8256067586552552935nteger @ ( minus_8373710615458151222nteger @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2963_power2__diff,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( power_2184487114949457152l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_2184487114949457152l_num1 @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7065122842183080059l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2964_power2__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( power_power_real @ ( minus_minus_real @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2965_power2__diff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2966_power2__diff,axiom,
    ! [X2: int,Y2: int] :
      ( ( power_power_int @ ( minus_minus_int @ X2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) ) ) ).

% power2_diff
thf(fact_2967_odd__0__le__power__imp__0__le,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2968_odd__0__le__power__imp__0__le,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2969_odd__0__le__power__imp__0__le,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2970_odd__0__le__power__imp__0__le,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_2971_odd__power__less__zero,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_z3403309356797280102nteger ) ) ).

% odd_power_less_zero
thf(fact_2972_odd__power__less__zero,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ord_less_real @ ( power_power_real @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).

% odd_power_less_zero
thf(fact_2973_odd__power__less__zero,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_2974_odd__power__less__zero,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ord_less_int @ ( power_power_int @ A3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).

% odd_power_less_zero
thf(fact_2975_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
thf(fact_2976_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
thf(fact_2977_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
thf(fact_2978_nat__div__eq__Suc__0__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( divide_divide_nat @ N2 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_div_eq_Suc_0_iff
thf(fact_2979_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
thf(fact_2980_field__sum__of__halves,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X2 ) ).

% field_sum_of_halves
thf(fact_2981_field__sum__of__halves,axiom,
    ! [X2: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X2 ) ).

% field_sum_of_halves
thf(fact_2982_arith__geo__mean,axiom,
    ! [U: real,X2: real,Y2: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X2 @ Y2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2983_arith__geo__mean,axiom,
    ! [U: rat,X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X2 @ Y2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2984_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_2985_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_2986_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_2987_power__2__mult__step__le,axiom,
    ! [N6: nat,N2: nat,K4: nat,K: nat] :
      ( ( ord_less_eq_nat @ N6 @ N2 )
     => ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ K4 ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ ( plus_plus_nat @ K4 @ one_one_nat ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% power_2_mult_step_le
thf(fact_2988_nat__less__power__trans,axiom,
    ! [N2: nat,M: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_less_power_trans
thf(fact_2989_sum__squares__bound,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_2990_sum__squares__bound,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y2 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_2991_nat__le__power__trans,axiom,
    ! [N2: nat,M: nat,K: nat] :
      ( ( ord_less_eq_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ K ) ) )
     => ( ( ord_less_eq_nat @ K @ M )
       => ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% nat_le_power_trans
thf(fact_2992_nat__power__less__diff,axiom,
    ! [N2: nat,Q2: nat,M: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Q2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     => ( ord_less_nat @ Q2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% nat_power_less_diff
thf(fact_2993_nat__add__offset__less,axiom,
    ! [Y2: nat,N2: nat,X2: nat,M: nat,Sz: nat] :
      ( ( ord_less_nat @ Y2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ( Sz
            = ( plus_plus_nat @ M @ N2 ) )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ Y2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Sz ) ) ) ) ) ).

% nat_add_offset_less
thf(fact_2994_two__powr__height__bound__deg,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_VEBT_height @ T2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% two_powr_height_bound_deg
thf(fact_2995_height__compose__summary,axiom,
    ! [Summary4: vEBT_VEBT,Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ Summary4 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) ) ) ).

% height_compose_summary
thf(fact_2996_height__compose__child,axiom,
    ! [T2: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Info4: option4927543243414619207at_nat,Deg4: nat,Summary4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( vEBT_VEBT_height @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ).

% height_compose_child
thf(fact_2997_delete__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% delete_bound_height'
thf(fact_2998_semiring__norm_I13_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% semiring_norm(13)
thf(fact_2999_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_3000_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times_num @ one @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_3001_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_3002_power__mult__numeral,axiom,
    ! [A3: nat,M: num,N2: num] :
      ( ( power_power_nat @ ( power_power_nat @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_3003_power__mult__numeral,axiom,
    ! [A3: int,M: num,N2: num] :
      ( ( power_power_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_3004_power__mult__numeral,axiom,
    ! [A3: real,M: num,N2: num] :
      ( ( power_power_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_3005_power__mult__numeral,axiom,
    ! [A3: complex,M: num,N2: num] :
      ( ( power_power_complex @ ( power_power_complex @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_3006_power__mult__numeral,axiom,
    ! [A3: code_integer,M: num,N2: num] :
      ( ( power_8256067586552552935nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_3007_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_3008_real__arch__pow__inv,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N4 ) @ Y2 ) ) ) ).

% real_arch_pow_inv
thf(fact_3009_real__arch__pow,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ? [N4: nat] : ( ord_less_real @ Y2 @ ( power_power_real @ X2 @ N4 ) ) ) ).

% real_arch_pow
thf(fact_3010_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3011_times__int__code_I2_J,axiom,
    ! [L2: int] :
      ( ( times_times_int @ zero_zero_int @ L2 )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3012_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_3013_four__x__squared,axiom,
    ! [X2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_3014_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A3 @ C ) ) @ ( times_times_real @ B3 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_3015_div__mult2__numeral__eq,axiom,
    ! [A3: nat,K: num,L2: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
      = ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_3016_div__mult2__numeral__eq,axiom,
    ! [A3: int,K: num,L2: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
      = ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_3017_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_3018_zdiv__mult__self,axiom,
    ! [M: int,A3: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ ( times_times_int @ M @ N2 ) ) @ M )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ M ) @ N2 ) ) ) ).

% zdiv_mult_self
thf(fact_3019_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_3020_realpow__pos__nth2,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ? [R3: real] :
          ( ( ord_less_real @ zero_zero_real @ R3 )
          & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
            = A3 ) ) ) ).

% realpow_pos_nth2
thf(fact_3021_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q5: int,R5: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R )
         => ( ( ord_less_int @ B3 @ R5 )
           => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3022_unique__quotient__lemma,axiom,
    ! [B3: int,Q5: int,R5: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
       => ( ( ord_less_int @ R5 @ B3 )
         => ( ( ord_less_int @ R @ B3 )
           => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3023_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q2: int,R: int,B4: int,Q5: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R5 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R5 ) @ zero_zero_int )
       => ( ( ord_less_int @ R @ B3 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B3 )
               => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3024_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q2: int,R: int,B4: int,Q5: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R5 ) )
       => ( ( ord_less_int @ R5 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B3 )
               => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3025_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q5 ) @ R5 ) )
     => ( ( ord_less_int @ R5 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).

% q_pos_lemma
thf(fact_3026_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3027_zdiv__zmult2__eq,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_3028_int__div__pos__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R: int] :
      ( ( A3
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B3 )
         => ( ( divide_divide_int @ A3 @ B3 )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3029_int__div__neg__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R: int] :
      ( ( A3
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R )
         => ( ( divide_divide_int @ A3 @ B3 )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3030_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 ) ) ) ) ) ).

% split_zdiv
thf(fact_3031_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R6: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R6 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R6 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_3032_z1pdiv2,axiom,
    ! [B3: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = B3 ) ).

% z1pdiv2
thf(fact_3033_pred__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% pred_bound_height'
thf(fact_3034_succ_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% succ'_bound_height
thf(fact_3035_member__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r2 @ T2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% member_bound_height'
thf(fact_3036_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N2 )
              = A3 )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A3 ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3037_realpow__pos__nth,axiom,
    ! [N2: nat,A3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ( ( power_power_real @ R3 @ N2 )
              = A3 ) ) ) ) ).

% realpow_pos_nth
thf(fact_3038_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ N2 )
     => ( ord_less_eq_nat @ one_one_nat @ ( vEBT_VEBT_height @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ).

% height_node
thf(fact_3039_diff__diff__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ M @ ( minus_minus_nat @ M @ N2 ) ) )
      = ( ( ord_less_nat @ I @ M )
        & ( ord_less_nat @ I @ N2 ) ) ) ).

% diff_diff_less
thf(fact_3040_neg__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
        = ( divide_divide_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A3 ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3041_pos__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
        = ( divide_divide_int @ B3 @ A3 ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3042_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3043_n__less__equal__power__2,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% n_less_equal_power_2
thf(fact_3044_msrevs_I1_J,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) @ N2 )
        = ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 ) @ K ) ) ) ).

% msrevs(1)
thf(fact_3045_nat__mult__power__less__eq,axiom,
    ! [B3: nat,A3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ B3 @ N2 ) ) @ ( power_power_nat @ B3 @ M ) )
        = ( ord_less_nat @ A3 @ ( power_power_nat @ B3 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% nat_mult_power_less_eq
thf(fact_3046_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R6 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R6 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_3047_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q4: int,R6: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R6 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R6 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3048_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q4: nat,R6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R6 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R6 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3049_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R6: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R6 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R6 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_def
thf(fact_3050_cnt__bound_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) @ one_one_real ) ) ) ) ).

% cnt_bound'
thf(fact_3051_real__average__minus__first,axiom,
    ! [A3: real,B3: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A3 )
      = ( divide_divide_real @ ( minus_minus_real @ B3 @ A3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_3052_real__average__minus__second,axiom,
    ! [B3: real,A3: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B3 @ A3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A3 )
      = ( divide_divide_real @ ( minus_minus_real @ B3 @ A3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_3053_bset_I6_J,axiom,
    ! [D4: int,B6: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B6 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ X4 @ T2 )
           => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T2 ) ) ) ) ).

% bset(6)
thf(fact_3054_bset_I8_J,axiom,
    ! [D4: int,T2: int,B6: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B6 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ T2 @ X4 )
             => ( ord_less_eq_int @ T2 @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_3055_aset_I6_J,axiom,
    ! [D4: int,T2: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A4 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ X4 @ T2 )
             => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(6)
thf(fact_3056_aset_I8_J,axiom,
    ! [D4: int,A4: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ T2 @ X4 )
           => ( ord_less_eq_int @ T2 @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_3057_pred__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_pred @ T2 @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T2 @ Y )
                & ( ord_less_nat @ Y @ X2 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_3058_cnt__non__neg,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq_real @ zero_zero_real @ ( vEBT_VEBT_cnt @ T2 ) ) ).

% cnt_non_neg
thf(fact_3059_mint__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_mint @ T2 )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T2 )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_3060_maxt__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T2 )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T2 )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_3061_succ__empty,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( ( vEBT_vebt_succ @ T2 @ X2 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y: nat] :
                ( ( vEBT_vebt_member @ T2 @ Y )
                & ( ord_less_nat @ X2 @ Y ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_3062_atLeastatMost__empty__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A3 @ B3 )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3063_atLeastatMost__empty__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( set_or633870826150836451st_rat @ A3 @ B3 )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3064_atLeastatMost__empty__iff,axiom,
    ! [A3: num,B3: num] :
      ( ( ( set_or7049704709247886629st_num @ A3 @ B3 )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3065_atLeastatMost__empty__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3066_atLeastatMost__empty__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ( set_or1266510415728281911st_int @ A3 @ B3 )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3067_atLeastatMost__empty__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( set_or1222579329274155063t_real @ A3 @ B3 )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_3068_atLeastatMost__empty__iff2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3069_atLeastatMost__empty__iff2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3070_atLeastatMost__empty__iff2,axiom,
    ! [A3: num,B3: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3071_atLeastatMost__empty__iff2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3072_atLeastatMost__empty__iff2,axiom,
    ! [A3: int,B3: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3073_atLeastatMost__empty__iff2,axiom,
    ! [A3: real,B3: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_3074_atLeastatMost__empty,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( set_or633870826150836451st_rat @ A3 @ B3 )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_3075_atLeastatMost__empty,axiom,
    ! [B3: num,A3: num] :
      ( ( ord_less_num @ B3 @ A3 )
     => ( ( set_or7049704709247886629st_num @ A3 @ B3 )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_3076_atLeastatMost__empty,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( ( set_or1269000886237332187st_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_3077_atLeastatMost__empty,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ B3 @ A3 )
     => ( ( set_or1266510415728281911st_int @ A3 @ B3 )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_3078_atLeastatMost__empty,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( set_or1222579329274155063t_real @ A3 @ B3 )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_3079_not__real__square__gt__zero,axiom,
    ! [X2: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
      = ( X2 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3080_minf_I7_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_real @ T2 @ X4 ) ) ).

% minf(7)
thf(fact_3081_minf_I7_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_rat @ T2 @ X4 ) ) ).

% minf(7)
thf(fact_3082_minf_I7_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_num @ T2 @ X4 ) ) ).

% minf(7)
thf(fact_3083_minf_I7_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_nat @ T2 @ X4 ) ) ).

% minf(7)
thf(fact_3084_minf_I7_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_int @ T2 @ X4 ) ) ).

% minf(7)
thf(fact_3085_minf_I5_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_real @ X4 @ T2 ) ) ).

% minf(5)
thf(fact_3086_minf_I5_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_rat @ X4 @ T2 ) ) ).

% minf(5)
thf(fact_3087_minf_I5_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_num @ X4 @ T2 ) ) ).

% minf(5)
thf(fact_3088_minf_I5_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_nat @ X4 @ T2 ) ) ).

% minf(5)
thf(fact_3089_minf_I5_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_int @ X4 @ T2 ) ) ).

% minf(5)
thf(fact_3090_minf_I4_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(4)
thf(fact_3091_minf_I4_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(4)
thf(fact_3092_minf_I4_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(4)
thf(fact_3093_minf_I4_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(4)
thf(fact_3094_minf_I4_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(4)
thf(fact_3095_minf_I3_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(3)
thf(fact_3096_minf_I3_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(3)
thf(fact_3097_minf_I3_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(3)
thf(fact_3098_minf_I3_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(3)
thf(fact_3099_minf_I3_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( X4 != T2 ) ) ).

% minf(3)
thf(fact_3100_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3101_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3102_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3103_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3104_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3105_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3106_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3107_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3108_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3109_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ X3 @ Z5 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3110_pinf_I7_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_real @ T2 @ X4 ) ) ).

% pinf(7)
thf(fact_3111_pinf_I7_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_rat @ T2 @ X4 ) ) ).

% pinf(7)
thf(fact_3112_pinf_I7_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_num @ T2 @ X4 ) ) ).

% pinf(7)
thf(fact_3113_pinf_I7_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_nat @ T2 @ X4 ) ) ).

% pinf(7)
thf(fact_3114_pinf_I7_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_int @ T2 @ X4 ) ) ).

% pinf(7)
thf(fact_3115_pinf_I5_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_real @ X4 @ T2 ) ) ).

% pinf(5)
thf(fact_3116_pinf_I5_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_rat @ X4 @ T2 ) ) ).

% pinf(5)
thf(fact_3117_pinf_I5_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_num @ X4 @ T2 ) ) ).

% pinf(5)
thf(fact_3118_pinf_I5_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T2 ) ) ).

% pinf(5)
thf(fact_3119_pinf_I5_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_int @ X4 @ T2 ) ) ).

% pinf(5)
thf(fact_3120_pinf_I4_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(4)
thf(fact_3121_pinf_I4_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(4)
thf(fact_3122_pinf_I4_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(4)
thf(fact_3123_pinf_I4_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(4)
thf(fact_3124_pinf_I4_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(4)
thf(fact_3125_pinf_I3_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(3)
thf(fact_3126_pinf_I3_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(3)
thf(fact_3127_pinf_I3_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(3)
thf(fact_3128_pinf_I3_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(3)
thf(fact_3129_pinf_I3_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( X4 != T2 ) ) ).

% pinf(3)
thf(fact_3130_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3131_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3132_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3133_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3134_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                | ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3135_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z5: real] :
        ! [X3: real] :
          ( ( ord_less_real @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: real] :
          ! [X3: real] :
            ( ( ord_less_real @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3136_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z5: rat] :
        ! [X3: rat] :
          ( ( ord_less_rat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: rat] :
          ! [X3: rat] :
            ( ( ord_less_rat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3137_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z5: num] :
        ! [X3: num] :
          ( ( ord_less_num @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: num] :
          ! [X3: num] :
            ( ( ord_less_num @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3138_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z5: nat] :
        ! [X3: nat] :
          ( ( ord_less_nat @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: nat] :
          ! [X3: nat] :
            ( ( ord_less_nat @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3139_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z5: int] :
        ! [X3: int] :
          ( ( ord_less_int @ Z5 @ X3 )
         => ( ( P @ X3 )
            = ( P6 @ X3 ) ) )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( Q @ X3 )
              = ( Q6 @ X3 ) ) )
       => ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
              = ( ( P6 @ X4 )
                & ( Q6 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3140_mult__commute__abs,axiom,
    ! [C: real] :
      ( ( ^ [X: real] : ( times_times_real @ X @ C ) )
      = ( times_times_real @ C ) ) ).

% mult_commute_abs
thf(fact_3141_mult__commute__abs,axiom,
    ! [C: rat] :
      ( ( ^ [X: rat] : ( times_times_rat @ X @ C ) )
      = ( times_times_rat @ C ) ) ).

% mult_commute_abs
thf(fact_3142_mult__commute__abs,axiom,
    ! [C: nat] :
      ( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
      = ( times_times_nat @ C ) ) ).

% mult_commute_abs
thf(fact_3143_mult__commute__abs,axiom,
    ! [C: int] :
      ( ( ^ [X: int] : ( times_times_int @ X @ C ) )
      = ( times_times_int @ C ) ) ).

% mult_commute_abs
thf(fact_3144_minf_I8_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ~ ( ord_less_eq_real @ T2 @ X4 ) ) ).

% minf(8)
thf(fact_3145_minf_I8_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ~ ( ord_less_eq_rat @ T2 @ X4 ) ) ).

% minf(8)
thf(fact_3146_minf_I8_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ~ ( ord_less_eq_num @ T2 @ X4 ) ) ).

% minf(8)
thf(fact_3147_minf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ~ ( ord_less_eq_nat @ T2 @ X4 ) ) ).

% minf(8)
thf(fact_3148_minf_I8_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ~ ( ord_less_eq_int @ T2 @ X4 ) ) ).

% minf(8)
thf(fact_3149_minf_I6_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ord_less_eq_real @ X4 @ T2 ) ) ).

% minf(6)
thf(fact_3150_minf_I6_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ord_less_eq_rat @ X4 @ T2 ) ) ).

% minf(6)
thf(fact_3151_minf_I6_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ X4 @ Z4 )
     => ( ord_less_eq_num @ X4 @ T2 ) ) ).

% minf(6)
thf(fact_3152_minf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ord_less_eq_nat @ X4 @ T2 ) ) ).

% minf(6)
thf(fact_3153_minf_I6_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ord_less_eq_int @ X4 @ T2 ) ) ).

% minf(6)
thf(fact_3154_pinf_I8_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ord_less_eq_real @ T2 @ X4 ) ) ).

% pinf(8)
thf(fact_3155_pinf_I8_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ord_less_eq_rat @ T2 @ X4 ) ) ).

% pinf(8)
thf(fact_3156_pinf_I8_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ( ord_less_eq_num @ T2 @ X4 ) ) ).

% pinf(8)
thf(fact_3157_pinf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ord_less_eq_nat @ T2 @ X4 ) ) ).

% pinf(8)
thf(fact_3158_pinf_I8_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ord_less_eq_int @ T2 @ X4 ) ) ).

% pinf(8)
thf(fact_3159_pinf_I6_J,axiom,
    ! [T2: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ~ ( ord_less_eq_real @ X4 @ T2 ) ) ).

% pinf(6)
thf(fact_3160_pinf_I6_J,axiom,
    ! [T2: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ~ ( ord_less_eq_rat @ X4 @ T2 ) ) ).

% pinf(6)
thf(fact_3161_pinf_I6_J,axiom,
    ! [T2: num] :
    ? [Z4: num] :
    ! [X4: num] :
      ( ( ord_less_num @ Z4 @ X4 )
     => ~ ( ord_less_eq_num @ X4 @ T2 ) ) ).

% pinf(6)
thf(fact_3162_pinf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T2 ) ) ).

% pinf(6)
thf(fact_3163_pinf_I6_J,axiom,
    ! [T2: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ~ ( ord_less_eq_int @ X4 @ T2 ) ) ).

% pinf(6)
thf(fact_3164_imp__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_3165_conj__le__cong,axiom,
    ! [X2: int,X7: int,P: $o,P6: $o] :
      ( ( X2 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_3166_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_12: int] : ( P6 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_3167_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_3168_bset_I1_J,axiom,
    ! [D4: int,B6: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B6 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B6 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                & ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(1)
thf(fact_3169_bset_I2_J,axiom,
    ! [D4: int,B6: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ B6 )
                 => ( X3
                   != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ B6 )
                   => ( X3
                     != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( minus_minus_int @ X4 @ D4 ) )
                | ( Q @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% bset(2)
thf(fact_3170_aset_I1_J,axiom,
    ! [D4: int,A4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                & ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                & ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(1)
thf(fact_3171_aset_I2_J,axiom,
    ! [D4: int,A4: set_int,P: int > $o,Q: int > $o] :
      ( ! [X3: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb3: int] :
                  ( ( member_int @ Xb3 @ A4 )
                 => ( X3
                   != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
         => ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
     => ( ! [X3: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb3: int] :
                    ( ( member_int @ Xb3 @ A4 )
                   => ( X3
                     != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
           => ( ( Q @ X3 )
             => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ( P @ X4 )
                | ( Q @ X4 ) )
             => ( ( P @ ( plus_plus_int @ X4 @ D4 ) )
                | ( Q @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ) ) ).

% aset(2)
thf(fact_3172_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3173_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P @ X3 )
           => ( P @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X4: int] :
              ( ( P @ X4 )
             => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3174_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K2: int] :
            ( ( P @ X3 )
            = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X6: int] : ( P @ X6 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_3175_aset_I7_J,axiom,
    ! [D4: int,A4: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ T2 @ X4 )
           => ( ord_less_int @ T2 @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_3176_aset_I5_J,axiom,
    ! [D4: int,T2: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ A4 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ X4 @ T2 )
             => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(5)
thf(fact_3177_aset_I4_J,axiom,
    ! [D4: int,T2: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ A4 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T2 )
             => ( ( plus_plus_int @ X4 @ D4 )
               != T2 ) ) ) ) ) ).

% aset(4)
thf(fact_3178_aset_I3_J,axiom,
    ! [D4: int,T2: int,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A4 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A4 )
                   => ( X4
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T2 )
             => ( ( plus_plus_int @ X4 @ D4 )
                = T2 ) ) ) ) ) ).

% aset(3)
thf(fact_3179_bset_I7_J,axiom,
    ! [D4: int,T2: int,B6: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ B6 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ T2 @ X4 )
             => ( ord_less_int @ T2 @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_3180_bset_I5_J,axiom,
    ! [D4: int,B6: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B6 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ X4 @ T2 )
           => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T2 ) ) ) ) ).

% bset(5)
thf(fact_3181_bset_I4_J,axiom,
    ! [D4: int,T2: int,B6: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ B6 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 != T2 )
             => ( ( minus_minus_int @ X4 @ D4 )
               != T2 ) ) ) ) ) ).

% bset(4)
thf(fact_3182_bset_I3_J,axiom,
    ! [D4: int,T2: int,B6: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B6 )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B6 )
                   => ( X4
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X4 = T2 )
             => ( ( minus_minus_int @ X4 @ D4 )
                = T2 ) ) ) ) ) ).

% bset(3)
thf(fact_3183_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A4 )
                     => ( X3
                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A4 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_3184_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B6: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B6 )
                     => ( X3
                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
             => ( ( P @ X3 )
               => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
         => ( ! [X3: int,K2: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X6: int] : ( P @ X6 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B6 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_3185_count__buildup,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% count_buildup
thf(fact_3186_cnt__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ T2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cnt_bound
thf(fact_3187_vebt__succ_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_succ @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( ( B
                     => ( Y2
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B
                     => ( Y2 = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_3188_psubsetI,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( A4 != B6 )
       => ( ord_less_set_nat @ A4 @ B6 ) ) ) ).

% psubsetI
thf(fact_3189_pos__mult__pos__ge,axiom,
    ! [X2: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ ( times_times_int @ N2 @ one_one_int ) @ ( times_times_int @ N2 @ X2 ) ) ) ) ).

% pos_mult_pos_ge
thf(fact_3190_vebt__pred_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: option_nat] :
      ( ( ( vEBT_vebt_pred @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A
                       => ( Y2 = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B
                           => ( Y2
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B
                           => ( ( A
                               => ( Y2
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A
                               => ( Y2 = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
                   => ( ( Y2 = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y2 = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2 = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_3191_set__bit__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ zero_zero_nat @ A3 )
      = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3192_set__bit__0,axiom,
    ! [A3: uint32] :
      ( ( bit_se6647067497041451410uint32 @ zero_zero_nat @ A3 )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3193_set__bit__0,axiom,
    ! [A3: int] :
      ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A3 )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3194_set__bit__0,axiom,
    ! [A3: nat] :
      ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A3 )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3195_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_min_max
thf(fact_3196_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X2 ) ).

% buildup_nothing_in_leaf
thf(fact_3197_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_3198_buildup__gives__valid,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).

% buildup_gives_valid
thf(fact_3199_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_3200_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_3201_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_3202_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_3203_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_3204_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_3205_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_3206_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_3207_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(73)
thf(fact_3208_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_3209_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_3210_semiring__norm_I14_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).

% semiring_norm(14)
thf(fact_3211_semiring__norm_I15_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).

% semiring_norm(15)
thf(fact_3212_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_3213_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(72)
thf(fact_3214_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_3215_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_3216_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_3217_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_3218_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_3219_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_3220_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_3221_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_3222_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_3223_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_3224_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_3225_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_3226_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_3227_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).

% set_bit_greater_eq
thf(fact_3228_xor__num_Ocases,axiom,
    ! [X2: product_prod_num_num] :
      ( ( X2
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N4: num] :
            ( X2
           != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
       => ( ! [N4: num] :
              ( X2
             != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
         => ( ! [M4: num] :
                ( X2
               != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
           => ( ! [M4: num,N4: num] :
                  ( X2
                 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
             => ( ! [M4: num,N4: num] :
                    ( X2
                   != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
               => ( ! [M4: num] :
                      ( X2
                     != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
                 => ( ! [M4: num,N4: num] :
                        ( X2
                       != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
                   => ~ ! [M4: num,N4: num] :
                          ( X2
                         != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_3229_num_Oexhaust,axiom,
    ! [Y2: num] :
      ( ( Y2 != one )
     => ( ! [X24: num] :
            ( Y2
           != ( bit0 @ X24 ) )
       => ~ ! [X33: num] :
              ( Y2
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_3230_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3231_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera9087168376688890119uint32 @ ( bit1 @ N2 ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) @ ( numera9087168376688890119uint32 @ N2 ) ) @ one_one_uint32 ) ) ).

% numeral_Bit1
thf(fact_3232_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit1 @ N2 ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ ( numera7442385471795722001l_num1 @ N2 ) ) @ one_on7727431528512463931l_num1 ) ) ).

% numeral_Bit1
thf(fact_3233_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_Bit1
thf(fact_3234_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_3235_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_Bit1
thf(fact_3236_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_Bit1
thf(fact_3237_eval__nat__numeral_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).

% eval_nat_numeral(3)
thf(fact_3238_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera9087168376688890119uint32 @ ( bit1 @ N2 ) )
      = ( plus_plus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) @ ( numera9087168376688890119uint32 @ N2 ) ) @ one_one_uint32 ) ) ).

% numeral_code(3)
thf(fact_3239_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numera7442385471795722001l_num1 @ ( bit1 @ N2 ) )
      = ( plus_p361126936061061375l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) @ ( numera7442385471795722001l_num1 @ N2 ) ) @ one_on7727431528512463931l_num1 ) ) ).

% numeral_code(3)
thf(fact_3240_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).

% numeral_code(3)
thf(fact_3241_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_3242_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).

% numeral_code(3)
thf(fact_3243_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).

% numeral_code(3)
thf(fact_3244_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_3245_power__numeral__odd,axiom,
    ! [Z: code_integer,W: num] :
      ( ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ Z @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_8256067586552552935nteger @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_3246_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_3247_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_3248_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_3249_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_3250_psubset__imp__ex__mem,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ B6 )
     => ? [B: vEBT_VEBT] : ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B6 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3251_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B6: set_int] :
      ( ( ord_less_set_int @ A4 @ B6 )
     => ? [B: int] : ( member_int @ B @ ( minus_minus_set_int @ B6 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3252_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B6: set_real] :
      ( ( ord_less_set_real @ A4 @ B6 )
     => ? [B: real] : ( member_real @ B @ ( minus_minus_set_real @ B6 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3253_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B6 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_3254_psubsetD,axiom,
    ! [A4: set_nat,B6: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_3255_psubsetD,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_VEBT_VEBT,C: vEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ B6 )
     => ( ( member_VEBT_VEBT @ C @ A4 )
       => ( member_VEBT_VEBT @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_3256_psubsetD,axiom,
    ! [A4: set_int,B6: set_int,C: int] :
      ( ( ord_less_set_int @ A4 @ B6 )
     => ( ( member_int @ C @ A4 )
       => ( member_int @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_3257_psubsetD,axiom,
    ! [A4: set_real,B6: set_real,C: real] :
      ( ( ord_less_set_real @ A4 @ B6 )
     => ( ( member_real @ C @ A4 )
       => ( member_real @ C @ B6 ) ) ) ).

% psubsetD
thf(fact_3258_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3259_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_3260_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_3261_power3__eq__cube,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3262_power3__eq__cube,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3263_power3__eq__cube,axiom,
    ! [A3: real] :
      ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3264_power3__eq__cube,axiom,
    ! [A3: rat] :
      ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_rat @ ( times_times_rat @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3265_power3__eq__cube,axiom,
    ! [A3: nat] :
      ( ( power_power_nat @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3266_power3__eq__cube,axiom,
    ! [A3: int] :
      ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A3 @ A3 ) @ A3 ) ) ).

% power3_eq_cube
thf(fact_3267_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_3268_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_3269_Set_Oempty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X: complex] : $false ) ) ).

% Set.empty_def
thf(fact_3270_Set_Oempty__def,axiom,
    ( bot_bo1796632182523588997nt_int
    = ( collec213857154873943460nt_int
      @ ^ [X: product_prod_int_int] : $false ) ) ).

% Set.empty_def
thf(fact_3271_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% Set.empty_def
thf(fact_3272_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_3273_Set_Oempty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X: int] : $false ) ) ).

% Set.empty_def
thf(fact_3274_num_Osize_I6_J,axiom,
    ! [X34: num] :
      ( ( size_size_num @ ( bit1 @ X34 ) )
      = ( plus_plus_nat @ ( size_size_num @ X34 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_3275_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A7 )
          @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_3276_less__set__def,axiom,
    ( ord_le3480810397992357184T_VEBT
    = ( ^ [A7: set_VEBT_VEBT,B7: set_VEBT_VEBT] :
          ( ord_less_VEBT_VEBT_o
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A7 )
          @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_3277_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A7 )
          @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_3278_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A7: set_real,B7: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A7 )
          @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ).

% less_set_def
thf(fact_3279_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_3280_not__psubset__empty,axiom,
    ! [A4: set_real] :
      ~ ( ord_less_set_real @ A4 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_3281_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_3282_not__psubset__empty,axiom,
    ! [A4: set_int] :
      ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_3283_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A7 @ B7 )
          | ( A7 = B7 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_3284_subset__psubset__trans,axiom,
    ! [A4: set_nat,B6: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B6 )
     => ( ( ord_less_set_nat @ B6 @ C4 )
       => ( ord_less_set_nat @ A4 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_3285_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A7 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_3286_psubset__subset__trans,axiom,
    ! [A4: set_nat,B6: set_nat,C4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ( ord_less_eq_set_nat @ B6 @ C4 )
       => ( ord_less_set_nat @ A4 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_3287_psubset__imp__subset,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ( ord_less_eq_set_nat @ A4 @ B6 ) ) ).

% psubset_imp_subset
thf(fact_3288_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A7 @ B7 )
          & ( A7 != B7 ) ) ) ) ).

% psubset_eq
thf(fact_3289_psubsetE,axiom,
    ! [A4: set_nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B6 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B6 )
         => ( ord_less_eq_set_nat @ B6 @ A4 ) ) ) ).

% psubsetE
thf(fact_3290_small__powers__of__2,axiom,
    ! [X2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ X2 )
     => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ X2 @ one_one_nat ) ) ) ) ).

% small_powers_of_2
thf(fact_3291_p2__eq__1,axiom,
    ! [N2: nat] :
      ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
        = one_on7727431528512463931l_num1 )
      = ( N2 = zero_zero_nat ) ) ).

% p2_eq_1
thf(fact_3292_word__less__two__pow__divD,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: nat,M: nat] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ ( divide1791077408188789448l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
        & ( ord_le750835935415966154l_num1 @ X2 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% word_less_two_pow_divD
thf(fact_3293_less__1__helper,axiom,
    ! [N2: int,M: int] :
      ( ( ord_less_eq_int @ N2 @ M )
     => ( ord_less_int @ ( minus_minus_int @ N2 @ one_one_int ) @ M ) ) ).

% less_1_helper
thf(fact_3294_space__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space @ T2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space_bound
thf(fact_3295_space_H__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_nat @ ( vEBT_VEBT_space2 @ T2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ U ) ) ) ) ).

% space'_bound
thf(fact_3296_delete__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% delete_bound_height
thf(fact_3297_tdeletemimi,axiom,
    ! [Deg4: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
     => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Mi ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% tdeletemimi
thf(fact_3298_vebt__buildup__bound,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ord_less_eq_nat @ ( vEBT_V8346862874174094_d_u_p @ N2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ U ) ) ) ).

% vebt_buildup_bound
thf(fact_3299_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y2 = one_one_nat ) )
                            & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                             => ( Y2
                                = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( if_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( plus_plus_nat @ ( vEBT_T_s_u_c_c2 @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                  @ one_one_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
thf(fact_3300_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y2 = one_one_nat ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( Y2
                                  = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( vEBT_T_p_r_e_d2 @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                                    @ one_one_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
thf(fact_3301_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( Xa = Mi2 ) @ zero_zero_nat
                              @ ( if_nat @ ( Xa = Ma2 ) @ zero_zero_nat
                                @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ zero_zero_nat
                                  @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ zero_zero_nat
                                    @ ( if_nat
                                      @ ( ( ord_less_nat @ Mi2 @ Xa )
                                        & ( ord_less_nat @ Xa @ Ma2 ) )
                                      @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( vEBT_T_m_e_m_b_e_r2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ zero_zero_nat )
                                      @ zero_zero_nat ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8099345112685741742_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
thf(fact_3302_space__space_H,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_nat @ ( vEBT_VEBT_space @ T2 ) @ ( vEBT_VEBT_space2 @ T2 ) ) ).

% space_space'
thf(fact_3303_word__coorder_Oextremum__unique,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = ( A3 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_unique
thf(fact_3304_word__le__0__iff,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ zero_z3563351764282998399l_num1 )
      = ( X2 = zero_z3563351764282998399l_num1 ) ) ).

% word_le_0_iff
thf(fact_3305_div__of__0__id,axiom,
    ! [N2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
      = zero_z3563351764282998399l_num1 ) ).

% div_of_0_id
thf(fact_3306_word__gt__0__no,axiom,
    ! [Y2: num] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( numera7442385471795722001l_num1 @ Y2 ) )
      = ( zero_z3563351764282998399l_num1
       != ( numera7442385471795722001l_num1 @ Y2 ) ) ) ).

% word_gt_0_no
thf(fact_3307_word__less__1,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ one_on7727431528512463931l_num1 )
      = ( X2 = zero_z3563351764282998399l_num1 ) ) ).

% word_less_1
thf(fact_3308_word__le__sub1__numberof,axiom,
    ! [W: num] :
      ( ( ( numera7442385471795722001l_num1 @ W )
       != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ W ) )
        = ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_sub1_numberof
thf(fact_3309_word__less__sub1__numberof,axiom,
    ! [W: num] :
      ( ( ( numera7442385471795722001l_num1 @ W )
       != zero_z3563351764282998399l_num1 )
     => ( ( ord_le750835935415966154l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ W ) )
        = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_sub1_numberof
thf(fact_3310_sub__wrap,axiom,
    ! [X2: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ X2 @ Z ) )
      = ( ( Z = zero_z3563351764282998399l_num1 )
        | ( ord_le750835935415966154l_num1 @ X2 @ Z ) ) ) ).

% sub_wrap
thf(fact_3311_less__1__simp,axiom,
    ! [N2: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ N2 @ one_on7727431528512463931l_num1 ) @ M )
      = ( ( ord_le3335648743751981014l_num1 @ N2 @ M )
        & ( N2 != zero_z3563351764282998399l_num1 ) ) ) ).

% less_1_simp
thf(fact_3312_le__m1__iff__lt,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X2 )
      = ( ( ord_le3335648743751981014l_num1 @ Y2 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) )
        = ( ord_le750835935415966154l_num1 @ Y2 @ X2 ) ) ) ).

% le_m1_iff_lt
thf(fact_3313_word__div__sub,axiom,
    ! [Y2: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ Y2 @ X2 )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ Y2 )
       => ( ( divide1791077408188789448l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ Y2 ) @ Y2 )
          = ( minus_4019991460397169231l_num1 @ ( divide1791077408188789448l_num1 @ X2 @ Y2 ) @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_div_sub
thf(fact_3314_word__less__nowrapI,axiom,
    ! [X2: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Z @ K ) )
     => ( ( ord_le3335648743751981014l_num1 @ K @ Z )
       => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ K )
         => ( ord_le750835935415966154l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ X2 @ K ) ) ) ) ) ).

% word_less_nowrapI
thf(fact_3315_word__diff__less,axiom,
    ! [N2: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ M )
       => ( ( ord_le3335648743751981014l_num1 @ N2 @ M )
         => ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ M @ N2 ) @ M ) ) ) ) ).

% word_diff_less
thf(fact_3316_plus__minus__not__NULL,axiom,
    ! [X2: word_N3645301735248828278l_num1,Ab2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Ab2 @ C ) )
     => ( ( ord_le3335648743751981014l_num1 @ C @ Ab2 )
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( ( plus_p361126936061061375l_num1 @ X2 @ C )
           != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL
thf(fact_3317_word__subset__less,axiom,
    ! [X2: word_N3645301735248828278l_num1,R: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
      ( ( ord_le5203802739334966412l_num1 @ ( set_or6221694504095523457l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ R ) @ one_on7727431528512463931l_num1 ) ) @ ( set_or6221694504095523457l_num1 @ Y2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ S ) @ one_on7727431528512463931l_num1 ) ) )
     => ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ X2 @ R ) @ one_on7727431528512463931l_num1 ) )
       => ( ( ord_le3335648743751981014l_num1 @ Y2 @ ( minus_4019991460397169231l_num1 @ ( plus_p361126936061061375l_num1 @ Y2 @ S ) @ one_on7727431528512463931l_num1 ) )
         => ( ( S != zero_z3563351764282998399l_num1 )
           => ( ord_le3335648743751981014l_num1 @ R @ S ) ) ) ) ) ).

% word_subset_less
thf(fact_3318_plus__minus__not__NULL__ab,axiom,
    ! [X2: word_N3645301735248828278l_num1,Ab2: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Ab2 @ C ) )
     => ( ( ord_le3335648743751981014l_num1 @ C @ Ab2 )
       => ( ( C != zero_z3563351764282998399l_num1 )
         => ( ( plus_p361126936061061375l_num1 @ X2 @ C )
           != zero_z3563351764282998399l_num1 ) ) ) ) ).

% plus_minus_not_NULL_ab
thf(fact_3319_word__less__nowrapI_H,axiom,
    ! [X2: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Z @ K ) )
     => ( ( ord_le3335648743751981014l_num1 @ K @ Z )
       => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ K )
         => ( ord_le750835935415966154l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ X2 @ K ) ) ) ) ) ).

% word_less_nowrapI'
thf(fact_3320_word__leq__le__minus__one,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ Y2 )
     => ( ( X2 != zero_z3563351764282998399l_num1 )
       => ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ Y2 ) ) ) ).

% word_leq_le_minus_one
thf(fact_3321_word__leq__minus__one__le,axiom,
    ! [Y2: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( Y2 != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Y2 @ one_on7727431528512463931l_num1 ) )
       => ( ord_le750835935415966154l_num1 @ X2 @ Y2 ) ) ) ).

% word_leq_minus_one_le
thf(fact_3322_word__sub__plus__one__nonzero,axiom,
    ! [N6: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ N6 @ N2 )
     => ( ( N6 != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ ( minus_4019991460397169231l_num1 @ N2 @ N6 ) @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% word_sub_plus_one_nonzero
thf(fact_3323_word__induct__less,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,M: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N4: word_N3645301735248828278l_num1] :
            ( ( ord_le750835935415966154l_num1 @ N4 @ M )
           => ( ( P @ N4 )
             => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N4 ) ) ) )
       => ( P @ M ) ) ) ).

% word_induct_less
thf(fact_3324_word__overflow,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ X2 @ one_on7727431528512463931l_num1 ) )
      | ( ( plus_p361126936061061375l_num1 @ X2 @ one_on7727431528512463931l_num1 )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_overflow
thf(fact_3325_word__gr0__conv__Suc,axiom,
    ! [M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ M )
     => ? [N4: word_N3645301735248828278l_num1] :
          ( M
          = ( plus_p361126936061061375l_num1 @ N4 @ one_on7727431528512463931l_num1 ) ) ) ).

% word_gr0_conv_Suc
thf(fact_3326_less__is__non__zero__p1,axiom,
    ! [A3: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ A3 @ K )
     => ( ( plus_p361126936061061375l_num1 @ A3 @ one_on7727431528512463931l_num1 )
       != zero_z3563351764282998399l_num1 ) ) ).

% less_is_non_zero_p1
thf(fact_3327_word__induct2,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,N2: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N4: word_N3645301735248828278l_num1] :
            ( ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N4 )
             != zero_z3563351764282998399l_num1 )
           => ( ( P @ N4 )
             => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N4 ) ) ) )
       => ( P @ N2 ) ) ) ).

% word_induct2
thf(fact_3328_word__induct,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,M: word_N3645301735248828278l_num1] :
      ( ( P @ zero_z3563351764282998399l_num1 )
     => ( ! [N4: word_N3645301735248828278l_num1] :
            ( ( P @ N4 )
           => ( P @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ N4 ) ) )
       => ( P @ M ) ) ) ).

% word_induct
thf(fact_3329_word__plus__one__nonzero,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ X2 @ Y2 ) )
     => ( ( Y2 != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ X2 @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% word_plus_one_nonzero
thf(fact_3330_plus__one__helper2,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ N2 )
     => ( ( ( plus_p361126936061061375l_num1 @ N2 @ one_on7727431528512463931l_num1 )
         != zero_z3563351764282998399l_num1 )
       => ( ord_le750835935415966154l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ N2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% plus_one_helper2
thf(fact_3331_neq__0__no__wrap,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( plus_p361126936061061375l_num1 @ X2 @ Y2 ) )
     => ( ( X2 != zero_z3563351764282998399l_num1 )
       => ( ( plus_p361126936061061375l_num1 @ X2 @ Y2 )
         != zero_z3563351764282998399l_num1 ) ) ) ).

% neq_0_no_wrap
thf(fact_3332_div__le__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ I @ ( divide1791077408188789448l_num1 @ K @ X2 ) )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X2 )
       => ( ord_le3335648743751981014l_num1 @ ( times_7065122842183080059l_num1 @ I @ X2 ) @ K ) ) ) ).

% div_le_mult
thf(fact_3333_div__lt__mult,axiom,
    ! [I: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ I @ ( divide1791077408188789448l_num1 @ K @ X2 ) )
     => ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X2 )
       => ( ord_le750835935415966154l_num1 @ ( times_7065122842183080059l_num1 @ I @ X2 ) @ K ) ) ) ).

% div_lt_mult
thf(fact_3334_More__Word_Oword__div__mult,axiom,
    ! [C: word_N3645301735248828278l_num1,A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ C )
     => ( ( ord_le750835935415966154l_num1 @ A3 @ ( times_7065122842183080059l_num1 @ B3 @ C ) )
       => ( ord_le750835935415966154l_num1 @ ( divide1791077408188789448l_num1 @ A3 @ C ) @ B3 ) ) ) ).

% More_Word.word_div_mult
thf(fact_3335_div__less__dividend__word,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( X2 != zero_z3563351764282998399l_num1 )
     => ( ( N2 != one_on7727431528512463931l_num1 )
       => ( ord_le750835935415966154l_num1 @ ( divide1791077408188789448l_num1 @ X2 @ N2 ) @ X2 ) ) ) ).

% div_less_dividend_word
thf(fact_3336_gt0__iff__gem1,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ X2 )
      = ( ord_le750835935415966154l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ X2 ) ) ).

% gt0_iff_gem1
thf(fact_3337_word__less__sub1,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( X2 != zero_z3563351764282998399l_num1 )
     => ( ( ord_le750835935415966154l_num1 @ one_on7727431528512463931l_num1 @ X2 )
        = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_less_sub1
thf(fact_3338_div__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
     => ( ( divide1791077408188789448l_num1 @ W @ W )
        = one_on7727431528512463931l_num1 ) ) ).

% div_word_self
thf(fact_3339_word__coorder_Oextremum,axiom,
    ! [A3: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ A3 ) ).

% word_coorder.extremum
thf(fact_3340_word__coorder_Oextremum__uniqueI,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
     => ( A3 = zero_z3563351764282998399l_num1 ) ) ).

% word_coorder.extremum_uniqueI
thf(fact_3341_word__zero__le,axiom,
    ! [Y2: word_N3645301735248828278l_num1] : ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ Y2 ) ).

% word_zero_le
thf(fact_3342_lt1__neq0,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X2 )
      = ( X2 != zero_z3563351764282998399l_num1 ) ) ).

% lt1_neq0
thf(fact_3343_word__must__wrap,axiom,
    ! [X2: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le3335648743751981014l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ N2 @ one_on7727431528512463931l_num1 ) )
     => ( ( ord_le3335648743751981014l_num1 @ N2 @ X2 )
       => ( N2 = zero_z3563351764282998399l_num1 ) ) ) ).

% word_must_wrap
thf(fact_3344_word__sub__1__le,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( X2 != zero_z3563351764282998399l_num1 )
     => ( ord_le3335648743751981014l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) @ X2 ) ) ).

% word_sub_1_le
thf(fact_3345_word__le__sub1,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( X2 != zero_z3563351764282998399l_num1 )
     => ( ( ord_le3335648743751981014l_num1 @ one_on7727431528512463931l_num1 @ X2 )
        = ( ord_le3335648743751981014l_num1 @ zero_z3563351764282998399l_num1 @ ( minus_4019991460397169231l_num1 @ X2 @ one_on7727431528512463931l_num1 ) ) ) ) ).

% word_le_sub1
thf(fact_3346_div__by__0__word,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ X2 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% div_by_0_word
thf(fact_3347_set__diff__eq,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A7: set_VEBT_VEBT,B7: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ A7 )
              & ~ ( member_VEBT_VEBT @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3348_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A7: set_real,B7: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A7 )
              & ~ ( member_real @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3349_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A7 )
              & ~ ( member_int @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3350_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A7: set_complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( member_complex @ X @ A7 )
              & ~ ( member_complex @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3351_set__diff__eq,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( member5262025264175285858nt_int @ X @ A7 )
              & ~ ( member5262025264175285858nt_int @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3352_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A7 )
              & ~ ( member_nat @ X @ B7 ) ) ) ) ) ).

% set_diff_eq
thf(fact_3353_minus__set__def,axiom,
    ( minus_5127226145743854075T_VEBT
    = ( ^ [A7: set_VEBT_VEBT,B7: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ( minus_2794559001203777698VEBT_o
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ A7 )
            @ ^ [X: vEBT_VEBT] : ( member_VEBT_VEBT @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3354_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A7: set_real,B7: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A7 )
            @ ^ [X: real] : ( member_real @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3355_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A7: set_int,B7: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A7 )
            @ ^ [X: int] : ( member_int @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3356_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A7: set_complex,B7: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X: complex] : ( member_complex @ X @ A7 )
            @ ^ [X: complex] : ( member_complex @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3357_minus__set__def,axiom,
    ( minus_1052850069191792384nt_int
    = ( ^ [A7: set_Pr958786334691620121nt_int,B7: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ( minus_711738161318947805_int_o
            @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A7 )
            @ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3358_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A7 )
            @ ^ [X: nat] : ( member_nat @ X @ B7 ) ) ) ) ) ).

% minus_set_def
thf(fact_3359_word__greater__zero__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = ( A3 != zero_z3563351764282998399l_num1 ) ) ).

% word_greater_zero_iff
thf(fact_3360_word__div__lt__eq__0,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X2 @ Y2 )
     => ( ( divide1791077408188789448l_num1 @ X2 @ Y2 )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_div_lt_eq_0
thf(fact_3361_word__gt__a__gt__0,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ A3 @ N2 )
     => ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N2 ) ) ).

% word_gt_a_gt_0
thf(fact_3362_word__less__div,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ X2 @ Y2 )
        = zero_z3563351764282998399l_num1 )
     => ( ( Y2 = zero_z3563351764282998399l_num1 )
        | ( ord_le750835935415966154l_num1 @ X2 @ Y2 ) ) ) ).

% word_less_div
thf(fact_3363_word__div__less,axiom,
    ! [W: word_N3645301735248828278l_num1,V: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ W @ V )
     => ( ( divide1791077408188789448l_num1 @ W @ V )
        = zero_z3563351764282998399l_num1 ) ) ).

% word_div_less
thf(fact_3364_word__neq__0__conv,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( W != zero_z3563351764282998399l_num1 )
      = ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ W ) ) ).

% word_neq_0_conv
thf(fact_3365_word__gt__0,axiom,
    ! [Y2: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ Y2 )
      = ( zero_z3563351764282998399l_num1 != Y2 ) ) ).

% word_gt_0
thf(fact_3366_word__coorder_Oextremum__strict,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ A3 @ zero_z3563351764282998399l_num1 ) ).

% word_coorder.extremum_strict
thf(fact_3367_word__not__simps_I1_J,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ~ ( ord_le750835935415966154l_num1 @ X2 @ zero_z3563351764282998399l_num1 ) ).

% word_not_simps(1)
thf(fact_3368_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
thf(fact_3369_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
thf(fact_3370_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg4 @ TreeList2 @ Summary4 ) @ Uu2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
thf(fact_3371_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
thf(fact_3372_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TreeList2 @ Summary4 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
thf(fact_3373_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ TreeList2 @ Summary4 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
thf(fact_3374_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% T_vebt_buildupi_univ
thf(fact_3375_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2
           != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ( ? [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
             => ( Y2
               != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => ( Y2
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
thf(fact_3376_vebt__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_vebt_member @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) @ Xa ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) )
                       => ( ( Xa != Mi2 )
                         => ( ( Xa != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3377_vebt__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_vebt_member @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
                 => ( ~ Y2
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y2
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( ( Xa != Mi2 )
                           => ( ( Xa != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3378_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3379_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3380_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Uu3: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu3 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy3: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y2
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy3 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3381_vebt__member_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_member @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) )
                 => ~ ( ( Xa != Mi2 )
                     => ( ( Xa != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_3382_T__vebt__buildupi__gq__0,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% T_vebt_buildupi_gq_0
thf(fact_3383_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
thf(fact_3384_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
thf(fact_3385_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
thf(fact_3386_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
thf(fact_3387_member__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ).

% member_bound_height
thf(fact_3388_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_m_e_m_b_e_r @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( X2 = Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = Ma ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
thf(fact_3389_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( vEBT_V441764108873111860ildupi @ N2 ) @ ( minus_minus_nat @ ( vEBT_VEBT_Tb2 @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi''
thf(fact_3390_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_m_e_m_b_e_r @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ( ! [V2: product_prod_nat_nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) )
                 => ( ( Y2
                      = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ( Y2
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( if_nat @ ( Xa = Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = Ma2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_e_m_b_e_r @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5837161174952499735_r_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
thf(fact_3391_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_membermima @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ~ Y2
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa ) ) ) )
         => ( ! [Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) )
               => ( ~ Y2
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) @ Xa ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
                 => ( ( Y2
                      = ( ( Xa = Mi2 )
                        | ( Xa = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y2
                        = ( ( Xa = Mi2 )
                          | ( Xa = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( Y2
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_3392_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa ) ) )
         => ( ! [Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux3 @ Uy3 ) @ Xa ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi2 )
                      | ( Xa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
                     => ( ( Xa = Mi2 )
                        | ( Xa = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_3393_space__2__pow__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( minus_minus_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) @ one_one_real ) ) ) ) ).

% space_2_pow_bound
thf(fact_3394_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Mi2: nat,Ma2: nat,Va4: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va4 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi2 )
                    | ( Xa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
                 => ~ ( ( Xa = Mi2 )
                      | ( Xa = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_3395_t__buildup__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8346862874174094_d_u_p @ N2 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_buildup_cnt
thf(fact_3396_T__vebt__buildupi__cnt_H,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V441764108873111860ildupi @ N2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) ) ) ).

% T_vebt_buildupi_cnt'
thf(fact_3397_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3398_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3399_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3400_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_3401_two__realpow__ge__two,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% two_realpow_ge_two
thf(fact_3402_count__buildup_H,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% count_buildup'
thf(fact_3403_space__cnt,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_space2 @ T2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_cnt @ T2 ) ) ) ).

% space_cnt
thf(fact_3404_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri8819519690708144855l_num1 @ zero_zero_nat )
    = zero_z3563351764282998399l_num1 ) ).

% semiring_1_class.of_nat_0
thf(fact_3405_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% semiring_1_class.of_nat_0
thf(fact_3406_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% semiring_1_class.of_nat_0
thf(fact_3407_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% semiring_1_class.of_nat_0
thf(fact_3408_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% semiring_1_class.of_nat_0
thf(fact_3409_semiring__1__class_Oof__nat__0,axiom,
    ( ( semiri4939895301339042750nteger @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% semiring_1_class.of_nat_0
thf(fact_3410_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3411_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3412_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3413_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3414_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_z3403309356797280102nteger
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_3415_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_3416_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_3417_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_3418_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_3419_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri4939895301339042750nteger @ M )
        = zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3420_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8819519690708144855l_num1 @ ( numeral_numeral_nat @ N2 ) )
      = ( numera7442385471795722001l_num1 @ N2 ) ) ).

% of_nat_numeral
thf(fact_3421_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% of_nat_numeral
thf(fact_3422_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_3423_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_3424_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_3425_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri4939895301339042750nteger @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% of_nat_numeral
thf(fact_3426_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3427_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3428_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3429_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3430_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_3431_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3432_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3433_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3434_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3435_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_3436_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3437_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_3438_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_3439_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_3440_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_add
thf(fact_3441_of__nat__1,axiom,
    ( ( semiri2565882477558803405uint32 @ one_one_nat )
    = one_one_uint32 ) ).

% of_nat_1
thf(fact_3442_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_3443_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_3444_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3445_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3446_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_3447_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3448_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_real
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3449_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_int
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3450_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_nat
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3451_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3452_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3453_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ N2 )
        = one_one_real )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3454_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ N2 )
        = one_one_int )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3455_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ N2 )
        = one_one_nat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3456_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ N2 )
        = one_one_Code_integer )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3457_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3458_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3459_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3460_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3461_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( times_times_nat @ M @ N2 ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mult
thf(fact_3462_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3463_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3464_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3465_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3466_semiring__1__class_Oof__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( power_power_nat @ M @ N2 ) )
      = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ M ) @ N2 ) ) ).

% semiring_1_class.of_nat_power
thf(fact_3467_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B3 ) @ W )
        = ( semiri8010041392384452111omplex @ X2 ) )
      = ( ( power_power_nat @ B3 @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3468_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W )
        = ( semiri5074537144036343181t_real @ X2 ) )
      = ( ( power_power_nat @ B3 @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3469_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W )
        = ( semiri1314217659103216013at_int @ X2 ) )
      = ( ( power_power_nat @ B3 @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3470_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W )
        = ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ( power_power_nat @ B3 @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3471_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W )
        = ( semiri4939895301339042750nteger @ X2 ) )
      = ( ( power_power_nat @ B3 @ W )
        = X2 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_3472_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X2 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B3 ) @ W ) )
      = ( X2
        = ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3473_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X2 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W ) )
      = ( X2
        = ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3474_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X2 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) )
      = ( X2
        = ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3475_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X2 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) )
      = ( X2
        = ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3476_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ( semiri4939895301339042750nteger @ X2 )
        = ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W ) )
      = ( X2
        = ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_3477_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_3478_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_3479_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_3480_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_3481_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_3482_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri2565882477558803405uint32 @ ( suc @ M ) )
      = ( plus_plus_uint32 @ one_one_uint32 @ ( semiri2565882477558803405uint32 @ M ) ) ) ).

% of_nat_Suc
thf(fact_3483_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3484_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_3485_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_3486_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_3487_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M ) ) ) ).

% of_nat_Suc
thf(fact_3488_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3489_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3490_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3491_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3492_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_3493_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 )
        = ( semiri8010041392384452111omplex @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3494_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 )
        = ( semiri681578069525770553at_rat @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3495_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 )
        = ( semiri5074537144036343181t_real @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3496_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = ( semiri1314217659103216013at_int @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3497_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = ( semiri1316708129612266289at_nat @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3498_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: nat] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 )
        = ( semiri4939895301339042750nteger @ Y2 ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_3499_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y2 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3500_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y2 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3501_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y2 )
        = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3502_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y2 )
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3503_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y2 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3504_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: nat,X2: num,N2: nat] :
      ( ( ( semiri4939895301339042750nteger @ Y2 )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_3505_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3506_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3507_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3508_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3509_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_3510_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3511_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3512_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3513_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3514_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_3515_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3516_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3517_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3518_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3519_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B3: nat,W: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W ) @ X2 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3520_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3521_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ B3 ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3522_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3523_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3524_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X2: nat,B3: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3525_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_3526_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_3527_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_3528_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3529_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3530_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3531_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3532_of__nat__zero__less__power__iff,axiom,
    ! [X2: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( semiri4939895301339042750nteger @ X2 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_3533_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3534_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3535_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3536_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3537_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N2 ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_3538_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3539_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3540_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3541_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3542_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N2 ) )
      = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_3543_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3544_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N2 ) @ ( semiri4939895301339042750nteger @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3545_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3546_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3547_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X2 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3548_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3549_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3550_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3551_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3552_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X2: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3553_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y2 )
      = ( times_times_rat @ Y2 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_3554_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y2 )
      = ( times_times_real @ Y2 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_3555_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y2 )
      = ( times_times_int @ Y2 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_3556_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y2 )
      = ( times_times_nat @ Y2 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_3557_mult__of__nat__commute,axiom,
    ! [X2: nat,Y2: code_integer] :
      ( ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ X2 ) @ Y2 )
      = ( times_3573771949741848930nteger @ Y2 @ ( semiri4939895301339042750nteger @ X2 ) ) ) ).

% mult_of_nat_commute
thf(fact_3558_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3559_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( semiri4939895301339042750nteger @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3560_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3561_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3562_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_3563_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3564_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_3565_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3566_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3567_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ zero_z3403309356797280102nteger ) ).

% of_nat_less_0_iff
thf(fact_3568_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3569_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3570_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3571_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3572_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ N2 ) )
     != zero_z3403309356797280102nteger ) ).

% semiring_char_0_class.of_nat_neq_0
thf(fact_3573_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3574_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3575_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3576_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3577_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_3578_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3579_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3580_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3581_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3582_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_3583_div__mult2__eq_H,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ A3 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_3584_div__mult2__eq_H,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ A3 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_3585_div__mult2__eq_H,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_3586_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_3587_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ I ) @ ( semiri4939895301339042750nteger @ J ) ) ) ).

% of_nat_mono
thf(fact_3588_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_3589_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_3590_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_3591_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3592_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3593_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_3594_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri8819519690708144855l_num1 @ K )
       != zero_z3563351764282998399l_num1 )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3595_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri681578069525770553at_rat @ K )
       != zero_zero_rat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3596_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri5074537144036343181t_real @ K )
       != zero_zero_real )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3597_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1314217659103216013at_int @ K )
       != zero_zero_int )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3598_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri1316708129612266289at_nat @ K )
       != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3599_of__nat__gt__0,axiom,
    ! [K: nat] :
      ( ( ( semiri4939895301339042750nteger @ K )
       != zero_z3403309356797280102nteger )
     => ( ord_less_nat @ zero_zero_nat @ K ) ) ).

% of_nat_gt_0
thf(fact_3600_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3601_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3602_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3603_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3604_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri4939895301339042750nteger @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_8373710615458151222nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_3605_reals__Archimedean3,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ! [Y4: real] :
        ? [N4: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X2 ) ) ) ).

% reals_Archimedean3
thf(fact_3606_real__of__nat__div4,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).

% real_of_nat_div4
thf(fact_3607_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).

% nat_less_real_le
thf(fact_3608_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3609_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3610_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3611_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3612_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_3613_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3614_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_3615_real__archimedian__rdiv__eq__0,axiom,
    ! [X2: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M4 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
         => ( X2 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3616_real__of__nat__div2,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) ) ) ).

% real_of_nat_div2
thf(fact_3617_real__of__nat__div3,axiom,
    ! [N2: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X2 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_3618_Tb_H__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( vEBT_VEBT_Tb2 @ N2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_cnt2 @ ( vEBT_vebt_buildup @ N2 ) ) ) ) ).

% Tb'_cnt
thf(fact_3619_cnt__cnt__eq,axiom,
    ( vEBT_VEBT_cnt
    = ( ^ [T: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( vEBT_VEBT_cnt2 @ T ) ) ) ) ).

% cnt_cnt_eq
thf(fact_3620_t__build__cnt,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V8646137997579335489_i_l_d @ N2 ) ) @ ( times_times_real @ ( vEBT_VEBT_cnt @ ( vEBT_vebt_buildup @ N2 ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ).

% t_build_cnt
thf(fact_3621_linear__plus__1__le__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_3622_delete__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% delete_bound_size_univ
thf(fact_3623_TBOUND__vebt__predi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% TBOUND_vebt_predi
thf(fact_3624_TBOUND__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_T8353473612707095248on_nat @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% TBOUND_vebt_succi
thf(fact_3625_buildup__build__time,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( vEBT_V8346862874174094_d_u_p @ N2 ) @ ( vEBT_V8646137997579335489_i_l_d @ N2 ) ) ).

% buildup_build_time
thf(fact_3626_TBOUND__vebt__minti,axiom,
    ! [T2: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_minti @ T2 ) @ one_one_nat ) ).

% TBOUND_vebt_minti
thf(fact_3627_TBOUND__vebt__maxti,axiom,
    ! [T2: vEBT_VEBTi] : ( time_T8353473612707095248on_nat @ ( vEBT_vebt_maxti @ T2 ) @ one_one_nat ) ).

% TBOUND_vebt_maxti
thf(fact_3628_TBOUND__replicate,axiom,
    ! [X2: heap_T2636463487746394924on_nat,C: nat,N2: nat] :
      ( ( time_T8353473612707095248on_nat @ X2 @ C )
     => ( time_T3808005469503390304on_nat @ ( vEBT_V792416675989592002on_nat @ N2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% TBOUND_replicate
thf(fact_3629_TBOUND__replicate,axiom,
    ! [X2: heap_Time_Heap_nat,C: nat,N2: nat] :
      ( ( time_TBOUND_nat @ X2 @ C )
     => ( time_TBOUND_list_nat @ ( vEBT_V7726092123322077554ei_nat @ N2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% TBOUND_replicate
thf(fact_3630_TBOUND__replicate,axiom,
    ! [X2: heap_Time_Heap_o,C: nat,N2: nat] :
      ( ( time_TBOUND_o @ X2 @ C )
     => ( time_TBOUND_list_o @ ( vEBT_V2326993469660664182atei_o @ N2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% TBOUND_replicate
thf(fact_3631_TBOUND__replicate,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,C: nat,N2: nat] :
      ( ( time_T5737551269749752165_VEBTi @ X2 @ C )
     => ( time_T8149879359713347829_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% TBOUND_replicate
thf(fact_3632_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_3633_delete__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% delete_bound_size_univ'
thf(fact_3634_height__double__log__univ__size,axiom,
    ! [U: real,Deg4: nat,T2: vEBT_VEBT] :
      ( ( U
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Deg4 ) )
     => ( ( vEBT_invar_vebt @ T2 @ Deg4 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_VEBT_height @ T2 ) ) @ ( plus_plus_real @ one_one_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% height_double_log_univ_size
thf(fact_3635_Abs__fnat__hom__0,axiom,
    ( zero_z3563351764282998399l_num1
    = ( semiri8819519690708144855l_num1 @ zero_zero_nat ) ) ).

% Abs_fnat_hom_0
thf(fact_3636_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_3637_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ? [N4: nat] :
          ( K
          = ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% zero_le_imp_eq_int
thf(fact_3638_nonneg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ~ ! [N4: nat] :
            ( K
           != ( semiri1314217659103216013at_int @ N4 ) ) ) ).

% nonneg_int_cases
thf(fact_3639_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W2: int,Z3: int] :
        ? [N: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_3640_zdiv__int,axiom,
    ! [A3: nat,B3: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A3 @ B3 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).

% zdiv_int
thf(fact_3641_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W2: int,Z3: int] :
        ? [N: nat] :
          ( Z3
          = ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3642_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ? [N4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N4 )
          & ( K
            = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3643_pos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ~ ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% pos_int_cases
thf(fact_3644_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_3645_zdiff__int__split,axiom,
    ! [P: int > $o,X2: nat,Y2: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y2 ) ) )
      = ( ( ( ord_less_eq_nat @ Y2 @ X2 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) )
        & ( ( ord_less_nat @ X2 @ Y2 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_3646_word__unat__power,axiom,
    ! [N2: nat] :
      ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
      = ( semiri8819519690708144855l_num1 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% word_unat_power
thf(fact_3647_pred__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d2 @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% pred_bound_size_univ'
thf(fact_3648_succ__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c2 @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ).

% succ_bound_size_univ'
thf(fact_3649_member__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_m_e_m_b_e_r @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% member_bound_size_univ
thf(fact_3650_Bolzano,axiom,
    ! [A3: real,B3: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ! [A: real,B: real,C2: real] :
            ( ( P @ A @ B )
           => ( ( P @ B @ C2 )
             => ( ( ord_less_eq_real @ A @ B )
               => ( ( ord_less_eq_real @ B @ C2 )
                 => ( P @ A @ C2 ) ) ) ) )
       => ( ! [X3: real] :
              ( ( ord_less_eq_real @ A3 @ X3 )
             => ( ( ord_less_eq_real @ X3 @ B3 )
               => ? [D5: real] :
                    ( ( ord_less_real @ zero_zero_real @ D5 )
                    & ! [A: real,B: real] :
                        ( ( ( ord_less_eq_real @ A @ X3 )
                          & ( ord_less_eq_real @ X3 @ B )
                          & ( ord_less_real @ ( minus_minus_real @ B @ A ) @ D5 ) )
                       => ( P @ A @ B ) ) ) ) )
         => ( P @ A3 @ B3 ) ) ) ) ).

% Bolzano
thf(fact_3651_log__pow__cancel,axiom,
    ! [A3: real,B3: nat] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( log @ A3 @ ( power_power_real @ A3 @ B3 ) )
          = ( semiri5074537144036343181t_real @ B3 ) ) ) ) ).

% log_pow_cancel
thf(fact_3652_Tb__T__vebt__buildupi,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N2 ) ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi
thf(fact_3653_zero__le__log__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A3 @ X2 ) )
          = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_3654_log__le__zero__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A3 @ X2 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_3655_one__le__log__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A3 @ X2 ) )
          = ( ord_less_eq_real @ A3 @ X2 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_3656_log__le__one__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ A3 @ X2 ) @ one_one_real )
          = ( ord_less_eq_real @ X2 @ A3 ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_3657_log__le__cancel__iff,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y2 ) )
            = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_3658_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_3659_TBOUND__highi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_highi @ X2 @ N2 ) @ one_one_nat ) ).

% TBOUND_highi
thf(fact_3660_TBOUND__lowi,axiom,
    ! [X2: nat,N2: nat] : ( time_TBOUND_nat @ ( vEBT_VEBT_lowi @ X2 @ N2 ) @ one_one_nat ) ).

% TBOUND_lowi
thf(fact_3661_Tb__Tb_H,axiom,
    ( vEBT_VEBT_Tb
    = ( ^ [T: nat] : ( semiri1314217659103216013at_int @ ( vEBT_VEBT_Tb2 @ T ) ) ) ) ).

% Tb_Tb'
thf(fact_3662_log__one,axiom,
    ! [A3: real] :
      ( ( log @ A3 @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_3663_zero__less__log__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A3 @ X2 ) )
          = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_3664_log__less__zero__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A3 @ X2 ) @ zero_zero_real )
          = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_3665_one__less__log__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A3 @ X2 ) )
          = ( ord_less_real @ A3 @ X2 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_3666_log__less__one__cancel__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ A3 @ X2 ) @ one_one_real )
          = ( ord_less_real @ X2 @ A3 ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_3667_log__less__cancel__iff,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y2 ) )
            = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_3668_log__eq__one,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( log @ A3 @ A3 )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_3669_TBOUND__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_TBOUND_o @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% TBOUND_vebt_memberi
thf(fact_3670_log__base__change,axiom,
    ! [A3: real,B3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( log @ B3 @ X2 )
          = ( divide_divide_real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ B3 ) ) ) ) ) ).

% log_base_change
thf(fact_3671_log__of__power__eq,axiom,
    ! [M: nat,B3: real,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B3 @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( semiri5074537144036343181t_real @ N2 )
          = ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_3672_less__log__of__power,axiom,
    ! [B3: real,N2: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B3 @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B3 @ M ) ) ) ) ).

% less_log_of_power
thf(fact_3673_log__mult,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A3 @ ( times_times_real @ X2 @ Y2 ) )
              = ( plus_plus_real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y2 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_3674_le__log__of__power,axiom,
    ! [B3: real,N2: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B3 @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B3 @ M ) ) ) ) ).

% le_log_of_power
thf(fact_3675_log__divide,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ zero_zero_real @ Y2 )
           => ( ( log @ A3 @ ( divide_divide_real @ X2 @ Y2 ) )
              = ( minus_minus_real @ ( log @ A3 @ X2 ) @ ( log @ A3 @ Y2 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_3676_log__base__pow,axiom,
    ! [A3: real,N2: nat,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( log @ ( power_power_real @ A3 @ N2 ) @ X2 )
        = ( divide_divide_real @ ( log @ A3 @ X2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_3677_log__nat__power,axiom,
    ! [X2: real,B3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ B3 @ ( power_power_real @ X2 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B3 @ X2 ) ) ) ) ).

% log_nat_power
thf(fact_3678_log2__of__power__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( semiri5074537144036343181t_real @ N2 )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_3679_log__of__power__less,axiom,
    ! [M: nat,B3: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_3680_log__of__power__le,axiom,
    ! [M: nat,B3: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_3681_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_3682_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_3683_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_3684_Tb__T__vebt__buildupi_H,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( vEBT_V9176841429113362141ildupi @ N2 ) @ ( minus_minus_int @ ( vEBT_VEBT_Tb @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% Tb_T_vebt_buildupi'
thf(fact_3685_TBOUND__buildupi,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_vebt_buildupi @ N2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% TBOUND_buildupi
thf(fact_3686_Tbuildupi__buildupi_H,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( vEBT_V441764108873111860ildupi @ N2 ) )
      = ( vEBT_V9176841429113362141ildupi @ N2 ) ) ).

% Tbuildupi_buildupi'
thf(fact_3687_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_3688_minNull__delete__time__bound,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
       => ( ord_less_eq_nat @ ( vEBT_T_d_e_l_e_t_e @ T2 @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% minNull_delete_time_bound
thf(fact_3689_VEBT__internal_OTb_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(2)
thf(fact_3690_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(2)
thf(fact_3691_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T2 )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ).

% not_min_Null_member
thf(fact_3692_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ~ ( vEBT_vebt_member @ T2 @ X2 ) ) ).

% min_Null_member
thf(fact_3693_minminNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T2 )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T2 ) ) ).

% minminNull
thf(fact_3694_minNullmin,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ( ( vEBT_vebt_mint @ T2 )
        = none_nat ) ) ).

% minNullmin
thf(fact_3695_minNull__delete__time__bound_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ T2 @ X2 ) )
       => ( ord_less_eq_nat @ ( vEBT_V1232361888498592333_e_t_e @ T2 @ X2 ) @ one_one_nat ) ) ) ).

% minNull_delete_time_bound'
thf(fact_3696_vebt__memberi__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine_Imp_refines_o @ ( vEBT_vebt_memberi @ Ti @ X2 ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 ) ) ).

% vebt_memberi_refines
thf(fact_3697_TBOUND__minNull,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_VEBT_minNull @ T2 )
     => ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ one_one_nat ) ) ).

% TBOUND_minNull
thf(fact_3698_TBOUND__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( if_nat @ ( vEBT_VEBT_minNull @ T2 ) @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ) ).

% TBOUND_vebt_inserti
thf(fact_3699_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ zero_zero_nat )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
thf(fact_3700_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_3701_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw3: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw3 @ Ux2 @ Uy2 ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_3702_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V9176841429113362141ildupi @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
thf(fact_3703_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu3: $o] :
              ( X2
             != ( vEBT_Leaf @ Uu3 @ $true ) )
         => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X2
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3704_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( X2
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
              ( X2
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3705_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y2 )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y2 )
         => ( ( ? [Uu3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu3 @ $true ) )
             => Y2 )
           => ( ( ? [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
               => ~ Y2 )
             => ~ ( ? [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                 => Y2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3706_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_T2636463487746394924on_nat,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F2: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3707_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_Time_Heap_o,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F2: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3708_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T2636463487746394924on_nat > heap_T8145700208782473153_VEBTi,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F2: $o > $o > heap_T2636463487746394924on_nat,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6250501799366334488on_nat @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3709_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_Time_Heap_o > heap_T2636463487746394924on_nat,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F2: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3710_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_Time_Heap_o > heap_Time_Heap_o,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F2: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3711_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_Time_Heap_o > heap_T8145700208782473153_VEBTi,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F2: $o > $o > heap_Time_Heap_o,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6104975476656191286Heap_o @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3712_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_T2636463487746394924on_nat,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6250501799366334488on_nat
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3713_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_Time_Heap_o,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6104975476656191286Heap_o
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3714_VEBTi_Ocase__distrib,axiom,
    ! [H2: heap_T8145700208782473153_VEBTi > heap_T8145700208782473153_VEBTi,F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,VEBTi: vEBT_VEBTi] :
      ( ( H2 @ ( vEBT_c6028912655521741485_VEBTi @ F1 @ F2 @ VEBTi ) )
      = ( vEBT_c6028912655521741485_VEBTi
        @ ^ [X12: option4927543243414619207at_nat,X23: nat,X32: array_VEBT_VEBTi,X42: vEBT_VEBTi] : ( H2 @ ( F1 @ X12 @ X23 @ X32 @ X42 ) )
        @ ^ [X12: $o,X23: $o] : ( H2 @ ( F2 @ X12 @ X23 ) )
        @ VEBTi ) ) ).

% VEBTi.case_distrib
thf(fact_3715_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ zero_zero_nat )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
thf(fact_3716_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_V441764108873111860ildupi @ ( suc @ zero_zero_nat ) )
    = ( suc @ zero_zero_nat ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
thf(fact_3717_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb2 @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb'.simps(1)
thf(fact_3718_VEBT__internal_OTb_Osimps_I1_J,axiom,
    ( ( vEBT_VEBT_Tb @ zero_zero_nat )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% VEBT_internal.Tb.simps(1)
thf(fact_3719_del__x__mia,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg4
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg4
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary4 ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ) ) ).

% del_x_mia
thf(fact_3720_del__x__mi__lets__in__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,Xn: nat,H2: nat,Summary4: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( 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 @ Summary4 @ H2 ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) )
                              @ Deg4
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_3721_del__x__mi__lets__in,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,Xn: nat,H2: nat,Summary4: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg4
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg4 @ Newlist @ Summary4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_3722_del__x__mi,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,Xn: nat,H2: nat,Summary4: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ 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 @ Summary4 @ H2 ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg4
                        @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                        @ ( vEBT_vebt_delete @ Summary4 @ 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 @ Deg4 @ ( 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 ) ) ) @ Deg4 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary4 ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_3723_del__in__range,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg4
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X2 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X2 != Mi )
                           => ( X2 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg4
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary4 ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) ) ) ) ) ) ).

% del_in_range
thf(fact_3724_del__x__not__mia,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg4: nat,H2: nat,L2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X2 = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ 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 @ Summary4 @ H2 ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg4
                      @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
                      @ ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) ) @ Deg4 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary4 ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_3725_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg4: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary4: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) )
                            @ Deg4
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_3726_vebt__inserti__refines,axiom,
    ! [Ti: vEBT_VEBTi,X2: nat,T2: vEBT_VEBT] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) ) ).

% vebt_inserti_refines
thf(fact_3727_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ~ ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ J @ X2 ) @ I )
        = ( nth_VEBT_VEBT @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3728_nth__update__invalid,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,J: nat,X2: vEBT_VEBTi] :
      ( ~ ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ J @ X2 ) @ I )
        = ( nth_VEBT_VEBTi @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3729_nth__update__invalid,axiom,
    ! [I: nat,L2: list_real,J: nat,X2: real] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( nth_real @ ( list_update_real @ L2 @ J @ X2 ) @ I )
        = ( nth_real @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3730_nth__update__invalid,axiom,
    ! [I: nat,L2: list_o,J: nat,X2: $o] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( nth_o @ ( list_update_o @ L2 @ J @ X2 ) @ I )
        = ( nth_o @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3731_nth__update__invalid,axiom,
    ! [I: nat,L2: list_nat,J: nat,X2: nat] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( nth_nat @ ( list_update_nat @ L2 @ J @ X2 ) @ I )
        = ( nth_nat @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3732_nth__update__invalid,axiom,
    ! [I: nat,L2: list_int,J: nat,X2: int] :
      ( ~ ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( nth_int @ ( list_update_int @ L2 @ J @ X2 ) @ I )
        = ( nth_int @ L2 @ I ) ) ) ).

% nth_update_invalid
thf(fact_3733_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg4: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg4 @ Newlist @ Summary4 ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_3734_del__x__mi__lets__in__not__minNull,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Deg4: nat,Xn: nat,H2: nat,Summary4: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X2 = Mi )
        & ( ord_less_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L2 )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg4 @ Newlist @ Summary4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_3735_del__x__not__mi,axiom,
    ! [Mi: nat,X2: nat,Ma: nat,Deg4: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X2 )
        & ( ord_less_eq_nat @ X2 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H2 )
           => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L2 )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( 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 ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X2 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg4
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary4 @ H2 ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg4 @ Newlist @ Summary4 ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_3736_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L2 ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L2 ) )
            & ! [Y: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3737_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L2 ) )
            & ! [Y: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3738_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L2 ) )
            & ! [Y: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3739_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_real,X2: real,Y2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_real @ X2 @ ( set_real2 @ L2 ) )
            & ! [Y: real] : ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3740_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_o,X2: $o,Y2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_o @ X2 @ ( set_o2 @ L2 ) )
            & ! [Y: $o] : ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3741_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_nat @ X2 @ ( set_nat2 @ L2 ) )
            & ! [Y: nat] : ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3742_in__set__upd__eq,axiom,
    ! [I: nat,L2: list_int,X2: int,Y2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ( ( member_int @ X2 @ ( set_int2 @ L2 ) )
            & ! [Y: int] : ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) ) ) ) ) ) ).

% in_set_upd_eq
thf(fact_3743_in__set__upd__cases,axiom,
    ! [X2: produc6575502325842934193n_assn,L2: list_P8527749157015355191n_assn,I: nat,Y2: produc6575502325842934193n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3744_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBT,L2: list_VEBT_VEBT,I: nat,Y2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3745_in__set__upd__cases,axiom,
    ! [X2: vEBT_VEBTi,L2: list_VEBT_VEBTi,I: nat,Y2: vEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3746_in__set__upd__cases,axiom,
    ! [X2: real,L2: list_real,I: nat,Y2: real] :
      ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member_real @ X2 @ ( set_real2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3747_in__set__upd__cases,axiom,
    ! [X2: $o,L2: list_o,I: nat,Y2: $o] :
      ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
         => ( X2 = ~ Y2 ) )
       => ( member_o @ X2 @ ( set_o2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3748_in__set__upd__cases,axiom,
    ! [X2: nat,L2: list_nat,I: nat,Y2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member_nat @ X2 @ ( set_nat2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3749_in__set__upd__cases,axiom,
    ! [X2: int,L2: list_int,I: nat,Y2: int] :
      ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) )
     => ( ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
         => ( X2 != Y2 ) )
       => ( member_int @ X2 @ ( set_int2 @ L2 ) ) ) ) ).

% in_set_upd_cases
thf(fact_3750_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L2 ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: produc6575502325842934193n_assn] : ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3751_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: vEBT_VEBT] : ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3752_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: vEBT_VEBTi] : ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3753_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_real,X2: real,Y2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: real] : ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3754_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_o,X2: $o,Y2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: $o] : ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3755_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: nat] : ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3756_in__set__upd__eq__aux,axiom,
    ! [I: nat,L2: list_int,X2: int,Y2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y2 ) ) )
        = ( ( X2 = Y2 )
          | ! [Y: int] : ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ L2 @ I @ Y ) ) ) ) ) ) ).

% in_set_upd_eq_aux
thf(fact_3757_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) ) )
       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X2 ) @ J )
          = ( nth_VEBT_VEBT @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3758_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) ) )
       => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X2 ) @ J )
          = ( nth_VEBT_VEBTi @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3759_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_real,X2: real] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) ) )
       => ( ( nth_real @ ( list_update_real @ L2 @ I @ X2 ) @ J )
          = ( nth_real @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3760_nth__list__update_H,axiom,
    ! [L2: list_o,I: nat,X2: $o,J: nat] :
      ( ( nth_o @ ( list_update_o @ L2 @ I @ X2 ) @ J )
      = ( ( ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => X2 )
        & ( ~ ( ( I = J )
              & ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) ) )
         => ( nth_o @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3761_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_nat,X2: nat] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) ) )
       => ( ( nth_nat @ ( list_update_nat @ L2 @ I @ X2 ) @ J )
          = ( nth_nat @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3762_nth__list__update_H,axiom,
    ! [I: nat,J: nat,L2: list_int,X2: int] :
      ( ( ( ( I = J )
          & ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) ) )
       => ( ( nth_int @ ( list_update_int @ L2 @ I @ X2 ) @ J )
          = X2 ) )
      & ( ~ ( ( I = J )
            & ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) ) )
       => ( ( nth_int @ ( list_update_int @ L2 @ I @ X2 ) @ J )
          = ( nth_int @ L2 @ J ) ) ) ) ).

% nth_list_update'
thf(fact_3763_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X2 @ Mi )
          | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = one_one_nat ) )
      & ( ~ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
              = one_one_nat ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_V1232361888498592333_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
              = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
thf(fact_3764_vebt__delete_Osimps_I7_J,axiom,
    ! [X2: nat,Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X2 @ Mi )
          | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) ) )
      & ( ~ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
       => ( ( ( ( X2 = Mi )
              & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) ) )
          & ( ~ ( ( X2 = Mi )
                & ( X2 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                            @ ( 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 @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X2 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X2 != Mi )
                             => ( X2 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( 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 @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va2 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary4 ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_3765_vebt__delete_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( ( Xa = zero_zero_nat )
             => ( Y2
               != ( vEBT_Leaf @ $false @ B ) ) ) )
       => ( ! [A: $o] :
              ( ? [B: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa
                  = ( suc @ zero_zero_nat ) )
               => ( Y2
                 != ( vEBT_Leaf @ A @ $false ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ? [N4: nat] :
                      ( Xa
                      = ( suc @ ( suc @ N4 ) ) )
                 => ( Y2
                   != ( vEBT_Leaf @ A @ B ) ) ) )
           => ( ! [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
                 => ( Y2
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y2
                     != ( 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] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y2
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y2
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_3766_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A: $o,B: $o] :
                ( X2
                = ( vEBT_Leaf @ A @ B ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ B ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( Y2 = one_one_nat ) )
                            & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                             => ( ( ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                 => ( Y2 = one_one_nat ) )
                                & ( ~ ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                 => ( Y2
                                    = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
thf(fact_3767_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_V1232361888498592333_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) )
                       => ( ( Y2 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y2 = one_one_nat ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y2 = one_one_nat ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( plus_plus_nat @ ( vEBT_V1232361888498592333_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_V1232361888498592333_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) ) @ one_one_nat ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V6368547301243506412_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
thf(fact_3768_vebt__delete_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2
                    = ( vEBT_Leaf @ $false @ B ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2
                      = ( vEBT_Leaf @ A @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2
                          = ( vEBT_Leaf @ A @ B ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
                   => ( ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y2
                          = ( 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 ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y2
                            = ( 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 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( ( ( ( ord_less_nat @ Xa @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( Y2
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                               => ( ( ( ( Xa = Mi2 )
                                      & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) )
                                  & ( ~ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                   => ( Y2
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa = Mi2 ) @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_3769_set__swap,axiom,
    ! [I: nat,Xs2: list_P8527749157015355191n_assn,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6829681357464350627n_assn @ Xs2 ) )
       => ( ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ ( list_u4534839942911652127n_assn @ Xs2 @ I @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ J ) ) @ J @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ I ) ) )
          = ( set_Pr1139785259514867910n_assn @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3770_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3771_set__swap,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
       => ( ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBTi @ Xs2 @ I ) ) )
          = ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3772_set__swap,axiom,
    ! [I: nat,Xs2: list_real,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ ( list_update_real @ Xs2 @ I @ ( nth_real @ Xs2 @ J ) ) @ J @ ( nth_real @ Xs2 @ I ) ) )
          = ( set_real2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3773_set__swap,axiom,
    ! [I: nat,Xs2: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3774_set__swap,axiom,
    ! [I: nat,Xs2: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3775_set__swap,axiom,
    ! [I: nat,Xs2: list_int,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_3776_insert__simp__excp,axiom,
    ! [Mi: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,X2: nat,Ma: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X2 @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg4 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary4 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary4 ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_3777_insert__simp__norm,axiom,
    ! [X2: nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary4: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X2 )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 )
         => ( ( X2 != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg4 @ TreeList2 @ Summary4 ) @ X2 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg4 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary4 ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_3778_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3779_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3780_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_real,X2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3781_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3782_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3783_nth__list__update__eq,axiom,
    ! [I: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ I )
        = X2 ) ) ).

% nth_list_update_eq
thf(fact_3784_TBOUND__minNulli,axiom,
    ! [T2: vEBT_VEBTi] : ( time_TBOUND_o @ ( vEBT_VEBT_minNulli @ T2 ) @ one_one_nat ) ).

% TBOUND_minNulli
thf(fact_3785_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_3786_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] :
      ( ( size_s7982070591426661849_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) )
      = ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ).

% length_list_update
thf(fact_3787_length__list__update,axiom,
    ! [Xs2: list_real,I: nat,X2: real] :
      ( ( size_size_list_real @ ( list_update_real @ Xs2 @ I @ X2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_list_update
thf(fact_3788_length__list__update,axiom,
    ! [Xs2: list_o,I: nat,X2: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I @ X2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_3789_length__list__update,axiom,
    ! [Xs2: list_nat,I: nat,X2: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_3790_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X2: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_3791_list__update__id,axiom,
    ! [Xs2: list_nat,I: nat] :
      ( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3792_list__update__id,axiom,
    ! [Xs2: list_int,I: nat] :
      ( ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3793_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat] :
      ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3794_list__update__id,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat] :
      ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ ( nth_VEBT_VEBTi @ Xs2 @ I ) )
      = Xs2 ) ).

% list_update_id
thf(fact_3795_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,X2: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
        = ( nth_nat @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3796_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_int,X2: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
        = ( nth_int @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3797_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
        = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3798_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) @ J )
        = ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ).

% nth_list_update_neq
thf(fact_3799_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_3800_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_3801_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_3802_max__nat_Oright__neutral,axiom,
    ! [A3: nat] :
      ( ( ord_max_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% max_nat.right_neutral
thf(fact_3803_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A3 @ B3 ) )
      = ( ( A3 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_3804_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% max_nat.left_neutral
thf(fact_3805_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ord_max_nat @ A3 @ B3 )
        = zero_zero_nat )
      = ( ( A3 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_3806_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(3)
thf(fact_3807_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X2 ) )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(3)
thf(fact_3808_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(3)
thf(fact_3809_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(3)
thf(fact_3810_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(3)
thf(fact_3811_max__0__1_I3_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(3)
thf(fact_3812_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(4)
thf(fact_3813_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ zero_z3403309356797280102nteger )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(4)
thf(fact_3814_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(4)
thf(fact_3815_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(4)
thf(fact_3816_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(4)
thf(fact_3817_max__0__1_I4_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(4)
thf(fact_3818_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_3819_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_3820_max__number__of_I1_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3335648743751981014l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( numera7442385471795722001l_num1 @ V ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( numera7442385471795722001l_num1 @ V ) )
          = ( numera7442385471795722001l_num1 @ V ) ) )
      & ( ~ ( ord_le3335648743751981014l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( numera7442385471795722001l_num1 @ V ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( numera7442385471795722001l_num1 @ V ) )
          = ( numera7442385471795722001l_num1 @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_3821_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_3822_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_3823_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_3824_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_3825_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_3826_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_3827_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_3828_max__0__1_I1_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(1)
thf(fact_3829_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_3830_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_3831_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_3832_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_3833_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_3834_max__0__1_I2_J,axiom,
    ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
    = one_on7984719198319812577d_enat ) ).

% max_0_1(2)
thf(fact_3835_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_3836_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_3837_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(5)
thf(fact_3838_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X2 ) )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(5)
thf(fact_3839_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X2 ) )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(5)
thf(fact_3840_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X2 ) )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(5)
thf(fact_3841_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(5)
thf(fact_3842_max__0__1_I5_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X2 ) )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(5)
thf(fact_3843_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X2 ) ) ).

% max_0_1(6)
thf(fact_3844_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X2 ) ) ).

% max_0_1(6)
thf(fact_3845_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ one_one_real )
      = ( numeral_numeral_real @ X2 ) ) ).

% max_0_1(6)
thf(fact_3846_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ X2 ) ) ).

% max_0_1(6)
thf(fact_3847_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat )
      = ( numeral_numeral_nat @ X2 ) ) ).

% max_0_1(6)
thf(fact_3848_max__0__1_I6_J,axiom,
    ! [X2: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ one_one_int )
      = ( numeral_numeral_int @ X2 ) ) ).

% max_0_1(6)
thf(fact_3849_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3850_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBTi,I: nat,X2: vEBT_VEBTi] :
      ( ( ord_less_eq_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ I )
     => ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3851_list__update__beyond,axiom,
    ! [Xs2: list_real,I: nat,X2: real] :
      ( ( ord_less_eq_nat @ ( size_size_list_real @ Xs2 ) @ I )
     => ( ( list_update_real @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3852_list__update__beyond,axiom,
    ! [Xs2: list_o,I: nat,X2: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
     => ( ( list_update_o @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3853_list__update__beyond,axiom,
    ! [Xs2: list_nat,I: nat,X2: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
     => ( ( list_update_nat @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3854_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X2: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X2 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_3855_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X2 ) @ ( semiri4216267220026989637d_enat @ Y2 ) ) ) ).

% of_nat_max
thf(fact_3856_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y2 ) ) ) ).

% of_nat_max
thf(fact_3857_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y2 ) ) ) ).

% of_nat_max
thf(fact_3858_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y2 ) ) ) ).

% of_nat_max
thf(fact_3859_of__nat__max,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( semiri4939895301339042750nteger @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ord_max_Code_integer @ ( semiri4939895301339042750nteger @ X2 ) @ ( semiri4939895301339042750nteger @ Y2 ) ) ) ).

% of_nat_max
thf(fact_3860_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_3861_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_3862_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_3863_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_3864_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_3865_neq__if__length__neq,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
       != ( size_size_list_real @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_3866_neq__if__length__neq,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
       != ( size_size_list_o @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_3867_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_3868_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_3869_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_real] :
      ( ( size_size_list_real @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3870_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3871_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3872_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_3873_length__induct,axiom,
    ! [P: list_real > $o,Xs2: list_real] :
      ( ! [Xs3: list_real] :
          ( ! [Ys2: list_real] :
              ( ( ord_less_nat @ ( size_size_list_real @ Ys2 ) @ ( size_size_list_real @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_3874_length__induct,axiom,
    ! [P: list_o > $o,Xs2: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_3875_length__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_3876_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_3877_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : Y5 = Z2 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I4 )
                = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3878_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBTi,Z2: list_VEBT_VEBTi] : Y5 = Z2 )
    = ( ^ [Xs: list_VEBT_VEBTi,Ys3: list_VEBT_VEBTi] :
          ( ( ( size_s7982070591426661849_VEBTi @ Xs )
            = ( size_s7982070591426661849_VEBTi @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s7982070591426661849_VEBTi @ Xs ) )
             => ( ( nth_VEBT_VEBTi @ Xs @ I4 )
                = ( nth_VEBT_VEBTi @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3879_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_real,Z2: list_real] : Y5 = Z2 )
    = ( ^ [Xs: list_real,Ys3: list_real] :
          ( ( ( size_size_list_real @ Xs )
            = ( size_size_list_real @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
             => ( ( nth_real @ Xs @ I4 )
                = ( nth_real @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3880_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z2: list_o] : Y5 = Z2 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I4 )
                = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3881_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z2: list_nat] : Y5 = Z2 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3882_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z2: list_int] : Y5 = Z2 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I4 )
                = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_3883_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3884_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBTi > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: vEBT_VEBTi] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_VEBT_VEBTi] :
            ( ( ( size_s7982070591426661849_VEBTi @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_VEBT_VEBTi @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3885_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > real > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: real] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_real] :
            ( ( ( size_size_list_real @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_real @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3886_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: $o] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3887_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: nat] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3888_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K )
           => ? [X6: int] : ( P @ I4 @ X6 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3889_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3890_nth__equalityI,axiom,
    ! [Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ( size_s7982070591426661849_VEBTi @ Xs2 )
        = ( size_s7982070591426661849_VEBTi @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( ( nth_VEBT_VEBTi @ Xs2 @ I2 )
              = ( nth_VEBT_VEBTi @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3891_nth__equalityI,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
           => ( ( nth_real @ Xs2 @ I2 )
              = ( nth_real @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3892_nth__equalityI,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
           => ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3893_nth__equalityI,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3894_nth__equalityI,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
           => ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Ys @ I2 ) ) )
       => ( Xs2 = Ys ) ) ) ).

% nth_equalityI
thf(fact_3895_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
           => ( P @ M3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_3896_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N2 )
            & ( P @ M3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_3897_length__pos__if__in__set,axiom,
    ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3898_length__pos__if__in__set,axiom,
    ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6829681357464350627n_assn @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3899_length__pos__if__in__set,axiom,
    ! [X2: real,Xs2: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3900_length__pos__if__in__set,axiom,
    ! [X2: $o,Xs2: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3901_length__pos__if__in__set,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3902_length__pos__if__in__set,axiom,
    ! [X2: int,Xs2: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_3903_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ! [X: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3904_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3905_all__set__conv__all__nth,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
           => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3906_all__set__conv__all__nth,axiom,
    ! [Xs2: list_real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
           => ( P @ ( nth_real @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3907_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3908_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3909_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
           => ( P @ X ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3910_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o,X2: vEBT_VEBTi] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3911_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
     => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3912_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o,X2: produc6575502325842934193n_assn] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
         => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ I2 ) ) )
     => ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3913_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P: real > $o,X2: real] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
         => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
     => ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3914_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P: $o > $o,X2: $o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
         => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
     => ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3915_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P: nat > $o,X2: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3916_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P: int > $o,X2: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
         => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
     => ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
       => ( P @ X2 ) ) ) ).

% all_nth_imp_all_set
thf(fact_3917_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBTi,Xs2: list_VEBT_VEBTi] :
      ( ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
            & ( ( nth_VEBT_VEBTi @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3918_in__set__conv__nth,axiom,
    ! [X2: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3919_in__set__conv__nth,axiom,
    ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
            & ( ( nth_Pr1769885009046257848n_assn @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3920_in__set__conv__nth,axiom,
    ! [X2: real,Xs2: list_real] :
      ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3921_in__set__conv__nth,axiom,
    ! [X2: $o,Xs2: list_o] :
      ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3922_in__set__conv__nth,axiom,
    ! [X2: nat,Xs2: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3923_in__set__conv__nth,axiom,
    ! [X2: int,Xs2: list_int] :
      ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I4 )
              = X2 ) ) ) ) ).

% in_set_conv_nth
thf(fact_3924_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,P: vEBT_VEBTi > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ! [X3: vEBT_VEBTi] :
            ( ( member_VEBT_VEBTi @ X3 @ ( set_VEBT_VEBTi2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3925_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3926_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_P8527749157015355191n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
     => ( ! [X3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3927_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_real,P: real > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_real @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3928_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_o @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3929_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3930_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P @ X3 ) )
       => ( P @ ( nth_int @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_3931_nth__mem,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) @ ( set_VEBT_VEBTi2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3932_nth__mem,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3933_nth__mem,axiom,
    ! [N2: nat,Xs2: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ N2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
     => ( member7957490590177025114n_assn @ ( nth_Pr1769885009046257848n_assn @ Xs2 @ N2 ) @ ( set_Pr1139785259514867910n_assn @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3934_nth__mem,axiom,
    ! [N2: nat,Xs2: list_real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ ( nth_real @ Xs2 @ N2 ) @ ( set_real2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3935_nth__mem,axiom,
    ! [N2: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ ( nth_o @ Xs2 @ N2 ) @ ( set_o2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3936_nth__mem,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ ( nth_nat @ Xs2 @ N2 ) @ ( set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3937_nth__mem,axiom,
    ! [N2: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ ( nth_int @ Xs2 @ N2 ) @ ( set_int2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_3938_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ N2 @ ( size_s6829681357464350627n_assn @ Xs2 ) )
     => ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3939_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3940_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( member_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3941_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_real,X2: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3942_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3943_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3944_set__update__memI,axiom,
    ! [N2: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs2 @ N2 @ X2 ) ) ) ) ).

% set_update_memI
thf(fact_3945_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3946_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_VEBT_VEBTi @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3947_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_real,X2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( list_update_real @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_real @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3948_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3949_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3950_list__update__same__conv,axiom,
    ! [I: nat,Xs2: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I @ X2 )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I )
          = X2 ) ) ) ).

% list_update_same_conv
thf(fact_3951_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X2 ) @ J )
            = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3952_nth__list__update,axiom,
    ! [I: nat,Xs2: list_VEBT_VEBTi,J: nat,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBTi @ ( list_u6098035379799741383_VEBTi @ Xs2 @ I @ X2 ) @ J )
            = ( nth_VEBT_VEBTi @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3953_nth__list__update,axiom,
    ! [I: nat,Xs2: list_real,J: nat,X2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_real @ ( list_update_real @ Xs2 @ I @ X2 ) @ J )
            = ( nth_real @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3954_nth__list__update,axiom,
    ! [I: nat,Xs2: list_o,X2: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X2 ) @ J )
        = ( ( ( I = J )
           => X2 )
          & ( ( I != J )
           => ( nth_o @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3955_nth__list__update,axiom,
    ! [I: nat,Xs2: list_nat,J: nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
            = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3956_nth__list__update,axiom,
    ! [I: nat,Xs2: list_int,J: nat,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
            = X2 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X2 ) @ J )
            = ( nth_int @ Xs2 @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_3957_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary4 ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3958_vebt__insert_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => ( Y2
                    = ( vEBT_Leaf @ $true @ B ) ) )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A @ $true ) ) )
                    & ( ( Xa != one_one_nat )
                     => ( Y2
                        = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
             => ( Y2
               != ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
               => ( Y2
                 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
           => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
                 => ( Y2
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3959_vebt__insert_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( ( Xa = zero_zero_nat )
                   => ( Y2
                      = ( vEBT_Leaf @ $true @ B ) ) )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A @ $true ) ) )
                      & ( ( Xa != one_one_nat )
                       => ( Y2
                          = ( vEBT_Leaf @ A @ B ) ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y2
                    = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y2
                      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
                   => ( ( Y2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3960_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2 != one_one_nat ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
               => ( Y2 != one_one_nat ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                     != ( if_nat
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                          & ~ ( ( Xa = Mi2 )
                              | ( Xa = Ma2 ) ) )
                        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                        @ one_one_nat ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
thf(fact_3961_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( if_nat
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X2 = Mi )
              | ( X2 = Ma ) ) )
        @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
        @ one_one_nat ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
thf(fact_3962_vebt__buildupi__refines,axiom,
    ! [N2: nat] : ( refine5565527176597971370_VEBTi @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_V739175172307565963ildupi @ N2 ) ) ).

% vebt_buildupi_refines
thf(fact_3963_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
thf(fact_3964_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ zero_zero_nat ) )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
thf(fact_3965_TBOUND__vebt__buildupi,axiom,
    ! [N2: nat] : ( time_T5737551269749752165_VEBTi @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% TBOUND_vebt_buildupi
thf(fact_3966_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = Q2 ) ).

% max_enat_simps(3)
thf(fact_3967_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = Q2 ) ).

% max_enat_simps(2)
thf(fact_3968_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
thf(fact_3969_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info4 @ zero_zero_nat @ Ts @ S ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
thf(fact_3970_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ Info4 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
thf(fact_3971_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2 @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
thf(fact_3972_insersimp_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ Y2 ) @ one_one_nat ) ) ) ).

% insersimp'
thf(fact_3973_insertsimp_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ T2 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ L2 ) @ one_one_nat ) ) ) ).

% insertsimp'
thf(fact_3974_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( X2
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X2
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
thf(fact_3975_insert_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t2 @ T2 @ X2 ) @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% insert'_bound_height
thf(fact_3976_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Leaf @ A3 @ B3 ) )
      = one_one_nat ) ).

% VEBT_internal.cnt'.simps(1)
thf(fact_3977_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    ( ( vEBT_V8646137997579335489_i_l_d @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
thf(fact_3978_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    ( ( vEBT_V8346862874174094_d_u_p @ zero_zero_nat )
    = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
thf(fact_3979_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Leaf @ A3 @ B3 ) )
      = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ).

% VEBT_internal.space'.simps(1)
thf(fact_3980_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_VEBT_space @ ( vEBT_Leaf @ A3 @ B3 ) )
      = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% VEBT_internal.space.simps(1)
thf(fact_3981_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t2 @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( if_nat
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                              & ~ ( ( Xa = Mi2 )
                                  | ( Xa = Ma2 ) ) )
                            @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t2 @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t2 @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                            @ one_one_nat ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T5076183648494686801_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
thf(fact_3982_T__vebt__buildupi,axiom,
    ! [N2: nat,H2: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ ( vEBT_V739175172307565963ildupi @ N2 ) @ H2 ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% T_vebt_buildupi
thf(fact_3983_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N4: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_3984_nat__approx__posE,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ~ ! [N4: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_3985_int__ops_I6_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A3 @ B3 ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).

% int_ops(6)
thf(fact_3986_max__less__iff__conj,axiom,
    ! [X2: extended_enat,Y2: extended_enat,Z: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) @ Z )
      = ( ( ord_le72135733267957522d_enat @ X2 @ Z )
        & ( ord_le72135733267957522d_enat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3987_max__less__iff__conj,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ X2 @ Y2 ) @ Z )
      = ( ( ord_le6747313008572928689nteger @ X2 @ Z )
        & ( ord_le6747313008572928689nteger @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3988_max__less__iff__conj,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( ord_less_real @ ( ord_max_real @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_real @ X2 @ Z )
        & ( ord_less_real @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3989_max__less__iff__conj,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_rat @ X2 @ Z )
        & ( ord_less_rat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3990_max__less__iff__conj,axiom,
    ! [X2: num,Y2: num,Z: num] :
      ( ( ord_less_num @ ( ord_max_num @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_num @ X2 @ Z )
        & ( ord_less_num @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3991_max__less__iff__conj,axiom,
    ! [X2: nat,Y2: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_nat @ X2 @ Z )
        & ( ord_less_nat @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3992_max__less__iff__conj,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X2 @ Y2 ) @ Z )
      = ( ( ord_less_int @ X2 @ Z )
        & ( ord_less_int @ Y2 @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_3993_max_Oabsorb4,axiom,
    ! [A3: extended_enat,B3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ A3 @ B3 )
     => ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3994_max_Oabsorb4,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
     => ( ( ord_max_Code_integer @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3995_max_Oabsorb4,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_max_real @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3996_max_Oabsorb4,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ( ord_max_rat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3997_max_Oabsorb4,axiom,
    ! [A3: num,B3: num] :
      ( ( ord_less_num @ A3 @ B3 )
     => ( ( ord_max_num @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3998_max_Oabsorb4,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_max_nat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_3999_max_Oabsorb4,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ( ord_max_int @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb4
thf(fact_4000_max_Oabsorb3,axiom,
    ! [B3: extended_enat,A3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
     => ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4001_max_Oabsorb3,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ B3 @ A3 )
     => ( ( ord_max_Code_integer @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4002_max_Oabsorb3,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( ( ord_max_real @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4003_max_Oabsorb3,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ B3 @ A3 )
     => ( ( ord_max_rat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4004_max_Oabsorb3,axiom,
    ! [B3: num,A3: num] :
      ( ( ord_less_num @ B3 @ A3 )
     => ( ( ord_max_num @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4005_max_Oabsorb3,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( ( ord_max_nat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4006_max_Oabsorb3,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ B3 @ A3 )
     => ( ( ord_max_int @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb3
thf(fact_4007_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
thf(fact_4008_time__replicate,axiom,
    ! [X2: heap_T5738788834812785303t_unit,C: nat,N2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ X2 @ H6 ) @ C )
     => ( ord_less_eq_nat @ ( time_t4781937132199089312t_unit @ ( vEBT_V7483891112628345579t_unit @ N2 @ X2 ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% time_replicate
thf(fact_4009_time__replicate,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,C: nat,N2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ X2 @ H6 ) @ C )
     => ( ord_less_eq_nat @ ( time_t3534373299052942712_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ N2 @ X2 ) @ H2 ) @ ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ C ) @ N2 ) ) ) ) ).

% time_replicate
thf(fact_4010_max_Oabsorb1,axiom,
    ! [B3: extended_enat,A3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
     => ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4011_max_Oabsorb1,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
     => ( ( ord_max_Code_integer @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4012_max_Oabsorb1,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( ord_max_rat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4013_max_Oabsorb1,axiom,
    ! [B3: num,A3: num] :
      ( ( ord_less_eq_num @ B3 @ A3 )
     => ( ( ord_max_num @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4014_max_Oabsorb1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( ord_max_nat @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4015_max_Oabsorb1,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( ord_max_int @ A3 @ B3 )
        = A3 ) ) ).

% max.absorb1
thf(fact_4016_max_Oabsorb2,axiom,
    ! [A3: extended_enat,B3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ A3 @ B3 )
     => ( ( ord_ma741700101516333627d_enat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4017_max_Oabsorb2,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
     => ( ( ord_max_Code_integer @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4018_max_Oabsorb2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_max_rat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4019_max_Oabsorb2,axiom,
    ! [A3: num,B3: num] :
      ( ( ord_less_eq_num @ A3 @ B3 )
     => ( ( ord_max_num @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4020_max_Oabsorb2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_max_nat @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4021_max_Oabsorb2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_max_int @ A3 @ B3 )
        = B3 ) ) ).

% max.absorb2
thf(fact_4022_max_Obounded__iff,axiom,
    ! [B3: extended_enat,C: extended_enat,A3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B3 @ C ) @ A3 )
      = ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
        & ( ord_le2932123472753598470d_enat @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4023_max_Obounded__iff,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
      = ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
        & ( ord_le3102999989581377725nteger @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4024_max_Obounded__iff,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
      = ( ( ord_less_eq_rat @ B3 @ A3 )
        & ( ord_less_eq_rat @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4025_max_Obounded__iff,axiom,
    ! [B3: num,C: num,A3: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 )
      = ( ( ord_less_eq_num @ B3 @ A3 )
        & ( ord_less_eq_num @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4026_max_Obounded__iff,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
      = ( ( ord_less_eq_nat @ B3 @ A3 )
        & ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4027_max_Obounded__iff,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 )
      = ( ( ord_less_eq_int @ B3 @ A3 )
        & ( ord_less_eq_int @ C @ A3 ) ) ) ).

% max.bounded_iff
thf(fact_4028_verit__la__disequality,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3 = B3 )
      | ~ ( ord_less_eq_rat @ A3 @ B3 )
      | ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_4029_verit__la__disequality,axiom,
    ! [A3: num,B3: num] :
      ( ( A3 = B3 )
      | ~ ( ord_less_eq_num @ A3 @ B3 )
      | ~ ( ord_less_eq_num @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_4030_verit__la__disequality,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 = B3 )
      | ~ ( ord_less_eq_nat @ A3 @ B3 )
      | ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_4031_verit__la__disequality,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 = B3 )
      | ~ ( ord_less_eq_int @ A3 @ B3 )
      | ~ ( ord_less_eq_int @ B3 @ A3 ) ) ).

% verit_la_disequality
thf(fact_4032_verit__comp__simplify1_I2_J,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_4033_verit__comp__simplify1_I2_J,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_4034_verit__comp__simplify1_I2_J,axiom,
    ! [A3: num] : ( ord_less_eq_num @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_4035_verit__comp__simplify1_I2_J,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_4036_verit__comp__simplify1_I2_J,axiom,
    ! [A3: int] : ( ord_less_eq_int @ A3 @ A3 ) ).

% verit_comp_simplify1(2)
thf(fact_4037_verit__comp__simplify1_I1_J,axiom,
    ! [A3: real] :
      ~ ( ord_less_real @ A3 @ A3 ) ).

% verit_comp_simplify1(1)
thf(fact_4038_verit__comp__simplify1_I1_J,axiom,
    ! [A3: rat] :
      ~ ( ord_less_rat @ A3 @ A3 ) ).

% verit_comp_simplify1(1)
thf(fact_4039_verit__comp__simplify1_I1_J,axiom,
    ! [A3: num] :
      ~ ( ord_less_num @ A3 @ A3 ) ).

% verit_comp_simplify1(1)
thf(fact_4040_verit__comp__simplify1_I1_J,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ A3 ) ).

% verit_comp_simplify1(1)
thf(fact_4041_verit__comp__simplify1_I1_J,axiom,
    ! [A3: int] :
      ~ ( ord_less_int @ A3 @ A3 ) ).

% verit_comp_simplify1(1)
thf(fact_4042_verit__la__generic,axiom,
    ! [A3: int,X2: int] :
      ( ( ord_less_eq_int @ A3 @ X2 )
      | ( A3 = X2 )
      | ( ord_less_eq_int @ X2 @ A3 ) ) ).

% verit_la_generic
thf(fact_4043_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A5: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A5 ) )
      = ( ord_less_real @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4044_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A5: rat] :
      ( ( ~ ( ord_less_eq_rat @ B4 @ A5 ) )
      = ( ord_less_rat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4045_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A5: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A5 ) )
      = ( ord_less_num @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4046_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A5: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
      = ( ord_less_nat @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4047_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A5: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
      = ( ord_less_int @ A5 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4048_verit__sum__simplify,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A3 @ zero_z3563351764282998399l_num1 )
      = A3 ) ).

% verit_sum_simplify
thf(fact_4049_verit__sum__simplify,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ A3 @ zero_zero_real )
      = A3 ) ).

% verit_sum_simplify
thf(fact_4050_verit__sum__simplify,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ A3 @ zero_zero_rat )
      = A3 ) ).

% verit_sum_simplify
thf(fact_4051_verit__sum__simplify,axiom,
    ! [A3: nat] :
      ( ( plus_plus_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% verit_sum_simplify
thf(fact_4052_verit__sum__simplify,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ A3 @ zero_zero_int )
      = A3 ) ).

% verit_sum_simplify
thf(fact_4053_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_4054_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X34: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X34 ) ) ).

% verit_eq_simplify(14)
thf(fact_4055_real__arch__simple,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% real_arch_simple
thf(fact_4056_real__arch__simple,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% real_arch_simple
thf(fact_4057_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N4: nat] :
            ( ~ ( P @ N4 )
            & ( P @ ( suc @ N4 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_4058_verit__eq__simplify_I12_J,axiom,
    ! [X34: num] :
      ( one
     != ( bit1 @ X34 ) ) ).

% verit_eq_simplify(12)
thf(fact_4059_reals__Archimedean2,axiom,
    ! [X2: rat] :
    ? [N4: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N4 ) ) ).

% reals_Archimedean2
thf(fact_4060_reals__Archimedean2,axiom,
    ! [X2: real] :
    ? [N4: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N4 ) ) ).

% reals_Archimedean2
thf(fact_4061_max_Omono,axiom,
    ! [C: extended_enat,A3: extended_enat,D: extended_enat,B3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A3 )
     => ( ( ord_le2932123472753598470d_enat @ D @ B3 )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4062_max_Omono,axiom,
    ! [C: code_integer,A3: code_integer,D: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A3 )
     => ( ( ord_le3102999989581377725nteger @ D @ B3 )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ C @ D ) @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4063_max_Omono,axiom,
    ! [C: rat,A3: rat,D: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A3 )
     => ( ( ord_less_eq_rat @ D @ B3 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4064_max_Omono,axiom,
    ! [C: num,A3: num,D: num,B3: num] :
      ( ( ord_less_eq_num @ C @ A3 )
     => ( ( ord_less_eq_num @ D @ B3 )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4065_max_Omono,axiom,
    ! [C: nat,A3: nat,D: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ( ord_less_eq_nat @ D @ B3 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4066_max_Omono,axiom,
    ! [C: int,A3: int,D: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A3 )
     => ( ( ord_less_eq_int @ D @ B3 )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A3 @ B3 ) ) ) ) ).

% max.mono
thf(fact_4067_max_OorderE,axiom,
    ! [B3: extended_enat,A3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
     => ( A3
        = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4068_max_OorderE,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
     => ( A3
        = ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4069_max_OorderE,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( A3
        = ( ord_max_rat @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4070_max_OorderE,axiom,
    ! [B3: num,A3: num] :
      ( ( ord_less_eq_num @ B3 @ A3 )
     => ( A3
        = ( ord_max_num @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4071_max_OorderE,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( A3
        = ( ord_max_nat @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4072_max_OorderE,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( A3
        = ( ord_max_int @ A3 @ B3 ) ) ) ).

% max.orderE
thf(fact_4073_max_OorderI,axiom,
    ! [A3: extended_enat,B3: extended_enat] :
      ( ( A3
        = ( ord_ma741700101516333627d_enat @ A3 @ B3 ) )
     => ( ord_le2932123472753598470d_enat @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4074_max_OorderI,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( A3
        = ( ord_max_Code_integer @ A3 @ B3 ) )
     => ( ord_le3102999989581377725nteger @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4075_max_OorderI,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3
        = ( ord_max_rat @ A3 @ B3 ) )
     => ( ord_less_eq_rat @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4076_max_OorderI,axiom,
    ! [A3: num,B3: num] :
      ( ( A3
        = ( ord_max_num @ A3 @ B3 ) )
     => ( ord_less_eq_num @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4077_max_OorderI,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3
        = ( ord_max_nat @ A3 @ B3 ) )
     => ( ord_less_eq_nat @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4078_max_OorderI,axiom,
    ! [A3: int,B3: int] :
      ( ( A3
        = ( ord_max_int @ A3 @ B3 ) )
     => ( ord_less_eq_int @ B3 @ A3 ) ) ).

% max.orderI
thf(fact_4079_max_OboundedE,axiom,
    ! [B3: extended_enat,C: extended_enat,A3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
         => ~ ( ord_le2932123472753598470d_enat @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4080_max_OboundedE,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
     => ~ ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
         => ~ ( ord_le3102999989581377725nteger @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4081_max_OboundedE,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_rat @ B3 @ A3 )
         => ~ ( ord_less_eq_rat @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4082_max_OboundedE,axiom,
    ! [B3: num,C: num,A3: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_num @ B3 @ A3 )
         => ~ ( ord_less_eq_num @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4083_max_OboundedE,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_nat @ B3 @ A3 )
         => ~ ( ord_less_eq_nat @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4084_max_OboundedE,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_eq_int @ B3 @ A3 )
         => ~ ( ord_less_eq_int @ C @ A3 ) ) ) ).

% max.boundedE
thf(fact_4085_max_OboundedI,axiom,
    ! [B3: extended_enat,A3: extended_enat,C: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ B3 @ A3 )
     => ( ( ord_le2932123472753598470d_enat @ C @ A3 )
       => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4086_max_OboundedI,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( ord_le3102999989581377725nteger @ B3 @ A3 )
     => ( ( ord_le3102999989581377725nteger @ C @ A3 )
       => ( ord_le3102999989581377725nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4087_max_OboundedI,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( ord_less_eq_rat @ B3 @ A3 )
     => ( ( ord_less_eq_rat @ C @ A3 )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4088_max_OboundedI,axiom,
    ! [B3: num,A3: num,C: num] :
      ( ( ord_less_eq_num @ B3 @ A3 )
     => ( ( ord_less_eq_num @ C @ A3 )
       => ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4089_max_OboundedI,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( ord_less_eq_nat @ C @ A3 )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4090_max_OboundedI,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( ord_less_eq_int @ B3 @ A3 )
     => ( ( ord_less_eq_int @ C @ A3 )
       => ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A3 ) ) ) ).

% max.boundedI
thf(fact_4091_max_Oorder__iff,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( A2
          = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4092_max_Oorder__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( A2
          = ( ord_max_Code_integer @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4093_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( A2
          = ( ord_max_rat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4094_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( A2
          = ( ord_max_num @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4095_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4096_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_4097_max_Ocobounded1,axiom,
    ! [A3: extended_enat,B3: extended_enat] : ( ord_le2932123472753598470d_enat @ A3 @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4098_max_Ocobounded1,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ A3 @ ( ord_max_Code_integer @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4099_max_Ocobounded1,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ A3 @ ( ord_max_rat @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4100_max_Ocobounded1,axiom,
    ! [A3: num,B3: num] : ( ord_less_eq_num @ A3 @ ( ord_max_num @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4101_max_Ocobounded1,axiom,
    ! [A3: nat,B3: nat] : ( ord_less_eq_nat @ A3 @ ( ord_max_nat @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4102_max_Ocobounded1,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ A3 @ ( ord_max_int @ A3 @ B3 ) ) ).

% max.cobounded1
thf(fact_4103_max_Ocobounded2,axiom,
    ! [B3: extended_enat,A3: extended_enat] : ( ord_le2932123472753598470d_enat @ B3 @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4104_max_Ocobounded2,axiom,
    ! [B3: code_integer,A3: code_integer] : ( ord_le3102999989581377725nteger @ B3 @ ( ord_max_Code_integer @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4105_max_Ocobounded2,axiom,
    ! [B3: rat,A3: rat] : ( ord_less_eq_rat @ B3 @ ( ord_max_rat @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4106_max_Ocobounded2,axiom,
    ! [B3: num,A3: num] : ( ord_less_eq_num @ B3 @ ( ord_max_num @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4107_max_Ocobounded2,axiom,
    ! [B3: nat,A3: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4108_max_Ocobounded2,axiom,
    ! [B3: int,A3: int] : ( ord_less_eq_int @ B3 @ ( ord_max_int @ A3 @ B3 ) ) ).

% max.cobounded2
thf(fact_4109_le__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) )
      = ( ( ord_le2932123472753598470d_enat @ Z @ X2 )
        | ( ord_le2932123472753598470d_enat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4110_le__max__iff__disj,axiom,
    ! [Z: code_integer,X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ Z @ ( ord_max_Code_integer @ X2 @ Y2 ) )
      = ( ( ord_le3102999989581377725nteger @ Z @ X2 )
        | ( ord_le3102999989581377725nteger @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4111_le__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y2 ) )
      = ( ( ord_less_eq_rat @ Z @ X2 )
        | ( ord_less_eq_rat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4112_le__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y2: num] :
      ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y2 ) )
      = ( ( ord_less_eq_num @ Z @ X2 )
        | ( ord_less_eq_num @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4113_le__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ( ord_less_eq_nat @ Z @ X2 )
        | ( ord_less_eq_nat @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4114_le__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ( ord_less_eq_int @ Z @ X2 )
        | ( ord_less_eq_int @ Z @ Y2 ) ) ) ).

% le_max_iff_disj
thf(fact_4115_max_Oabsorb__iff1,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4116_max_Oabsorb__iff1,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4117_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4118_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4119_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4120_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_4121_max_Oabsorb__iff2,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] :
          ( ( ord_ma741700101516333627d_enat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4122_max_Oabsorb__iff2,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ord_max_Code_integer @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4123_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( ord_max_rat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4124_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A2: num,B2: num] :
          ( ( ord_max_num @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4125_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4126_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_4127_max_OcoboundedI1,axiom,
    ! [C: extended_enat,A3: extended_enat,B3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ A3 )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4128_max_OcoboundedI1,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ A3 )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4129_max_OcoboundedI1,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ C @ A3 )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4130_max_OcoboundedI1,axiom,
    ! [C: num,A3: num,B3: num] :
      ( ( ord_less_eq_num @ C @ A3 )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4131_max_OcoboundedI1,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ C @ A3 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4132_max_OcoboundedI1,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ C @ A3 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).

% max.coboundedI1
thf(fact_4133_max_OcoboundedI2,axiom,
    ! [C: extended_enat,B3: extended_enat,A3: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ C @ B3 )
     => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4134_max_OcoboundedI2,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C @ B3 )
     => ( ord_le3102999989581377725nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4135_max_OcoboundedI2,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ C @ B3 )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4136_max_OcoboundedI2,axiom,
    ! [C: num,B3: num,A3: num] :
      ( ( ord_less_eq_num @ C @ B3 )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4137_max_OcoboundedI2,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ C @ B3 )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4138_max_OcoboundedI2,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( ord_less_eq_int @ C @ B3 )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).

% max.coboundedI2
thf(fact_4139_max_Ostrict__coboundedI2,axiom,
    ! [C: extended_enat,B3: extended_enat,A3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ B3 )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4140_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ B3 )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4141_max_Ostrict__coboundedI2,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ C @ B3 )
     => ( ord_less_real @ C @ ( ord_max_real @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4142_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ C @ B3 )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4143_max_Ostrict__coboundedI2,axiom,
    ! [C: num,B3: num,A3: num] :
      ( ( ord_less_num @ C @ B3 )
     => ( ord_less_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4144_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( ord_less_nat @ C @ B3 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4145_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( ord_less_int @ C @ B3 )
     => ( ord_less_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI2
thf(fact_4146_max_Ostrict__coboundedI1,axiom,
    ! [C: extended_enat,A3: extended_enat,B3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ C @ A3 )
     => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4147_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C @ A3 )
     => ( ord_le6747313008572928689nteger @ C @ ( ord_max_Code_integer @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4148_max_Ostrict__coboundedI1,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ A3 )
     => ( ord_less_real @ C @ ( ord_max_real @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4149_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ A3 )
     => ( ord_less_rat @ C @ ( ord_max_rat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4150_max_Ostrict__coboundedI1,axiom,
    ! [C: num,A3: num,B3: num] :
      ( ( ord_less_num @ C @ A3 )
     => ( ord_less_num @ C @ ( ord_max_num @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4151_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ C @ A3 )
     => ( ord_less_nat @ C @ ( ord_max_nat @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4152_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_int @ C @ A3 )
     => ( ord_less_int @ C @ ( ord_max_int @ A3 @ B3 ) ) ) ).

% max.strict_coboundedI1
thf(fact_4153_max_Ostrict__order__iff,axiom,
    ( ord_le72135733267957522d_enat
    = ( ^ [B2: extended_enat,A2: extended_enat] :
          ( ( A2
            = ( ord_ma741700101516333627d_enat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4154_max_Ostrict__order__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [B2: code_integer,A2: code_integer] :
          ( ( A2
            = ( ord_max_Code_integer @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4155_max_Ostrict__order__iff,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A2: real] :
          ( ( A2
            = ( ord_max_real @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4156_max_Ostrict__order__iff,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A2: rat] :
          ( ( A2
            = ( ord_max_rat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4157_max_Ostrict__order__iff,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A2: num] :
          ( ( A2
            = ( ord_max_num @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4158_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4159_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_4160_max_Ostrict__boundedE,axiom,
    ! [B3: extended_enat,C: extended_enat,A3: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_le72135733267957522d_enat @ B3 @ A3 )
         => ~ ( ord_le72135733267957522d_enat @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4161_max_Ostrict__boundedE,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( ord_max_Code_integer @ B3 @ C ) @ A3 )
     => ~ ( ( ord_le6747313008572928689nteger @ B3 @ A3 )
         => ~ ( ord_le6747313008572928689nteger @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4162_max_Ostrict__boundedE,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ord_less_real @ ( ord_max_real @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_real @ B3 @ A3 )
         => ~ ( ord_less_real @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4163_max_Ostrict__boundedE,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_rat @ ( ord_max_rat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_rat @ B3 @ A3 )
         => ~ ( ord_less_rat @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4164_max_Ostrict__boundedE,axiom,
    ! [B3: num,C: num,A3: num] :
      ( ( ord_less_num @ ( ord_max_num @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_num @ B3 @ A3 )
         => ~ ( ord_less_num @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4165_max_Ostrict__boundedE,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_nat @ B3 @ A3 )
         => ~ ( ord_less_nat @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4166_max_Ostrict__boundedE,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( ord_less_int @ ( ord_max_int @ B3 @ C ) @ A3 )
     => ~ ( ( ord_less_int @ B3 @ A3 )
         => ~ ( ord_less_int @ C @ A3 ) ) ) ).

% max.strict_boundedE
thf(fact_4167_less__max__iff__disj,axiom,
    ! [Z: extended_enat,X2: extended_enat,Y2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X2 @ Y2 ) )
      = ( ( ord_le72135733267957522d_enat @ Z @ X2 )
        | ( ord_le72135733267957522d_enat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4168_less__max__iff__disj,axiom,
    ! [Z: code_integer,X2: code_integer,Y2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z @ ( ord_max_Code_integer @ X2 @ Y2 ) )
      = ( ( ord_le6747313008572928689nteger @ Z @ X2 )
        | ( ord_le6747313008572928689nteger @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4169_less__max__iff__disj,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y2 ) )
      = ( ( ord_less_real @ Z @ X2 )
        | ( ord_less_real @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4170_less__max__iff__disj,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y2 ) )
      = ( ( ord_less_rat @ Z @ X2 )
        | ( ord_less_rat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4171_less__max__iff__disj,axiom,
    ! [Z: num,X2: num,Y2: num] :
      ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y2 ) )
      = ( ( ord_less_num @ Z @ X2 )
        | ( ord_less_num @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4172_less__max__iff__disj,axiom,
    ! [Z: nat,X2: nat,Y2: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ Z @ X2 )
        | ( ord_less_nat @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4173_less__max__iff__disj,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y2 ) )
      = ( ( ord_less_int @ Z @ X2 )
        | ( ord_less_int @ Z @ Y2 ) ) ) ).

% less_max_iff_disj
thf(fact_4174_max__def__raw,axiom,
    ( ord_ma741700101516333627d_enat
    = ( ^ [A2: extended_enat,B2: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4175_max__def__raw,axiom,
    ( ord_max_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] : ( if_Code_integer @ ( ord_le3102999989581377725nteger @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4176_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4177_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A2: rat,B2: rat] : ( if_rat @ ( ord_less_eq_rat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4178_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4179_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4180_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def_raw
thf(fact_4181_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_4182_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_4183_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_4184_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_4185_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_4186_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_4187_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4188_ex__less__of__nat__mult,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X2 )
     => ? [N4: nat] : ( ord_less_rat @ Y2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4189_ex__less__of__nat__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [N4: nat] : ( ord_less_real @ Y2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X2 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_4190_int__ops_I4_J,axiom,
    ! [A3: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ A3 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ).

% int_ops(4)
thf(fact_4191_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_4192_heigt__uplog__rel,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ T2 ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% heigt_uplog_rel
thf(fact_4193_succ__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% succ_bound_size_univ
thf(fact_4194_TBOUND__option__case,axiom,
    ! [T2: option_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: nat > heap_Time_Heap_nat,Bnd2: nat > nat] :
      ( ( ( T2 = none_nat )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T2
              = ( some_nat @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o6609685678014844897at_nat @ F @ F3 @ T2 ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4195_TBOUND__option__case,axiom,
    ! [T2: option_num,F: heap_Time_Heap_nat,Bnd: nat,F3: num > heap_Time_Heap_nat,Bnd2: num > nat] :
      ( ( ( T2 = none_num )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T2
              = ( some_num @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o3167017464170623531at_num @ F @ F3 @ T2 ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4196_TBOUND__option__case,axiom,
    ! [T2: option_nat,F: heap_Time_Heap_o,Bnd: nat,F3: nat > heap_Time_Heap_o,Bnd2: nat > nat] :
      ( ( ( T2 = none_nat )
       => ( time_TBOUND_o @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T2
              = ( some_nat @ X3 ) )
           => ( time_TBOUND_o @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_o @ ( case_o6892868863119666303_o_nat @ F @ F3 @ T2 ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4197_TBOUND__option__case,axiom,
    ! [T2: option_num,F: heap_Time_Heap_o,Bnd: nat,F3: num > heap_Time_Heap_o,Bnd2: num > nat] :
      ( ( ( T2 = none_num )
       => ( time_TBOUND_o @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T2
              = ( some_num @ X3 ) )
           => ( time_TBOUND_o @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_o @ ( case_o3450200649275444937_o_num @ F @ F3 @ T2 ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4198_TBOUND__option__case,axiom,
    ! [T2: option_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: nat > heap_T8145700208782473153_VEBTi,Bnd2: nat > nat] :
      ( ( ( T2 = none_nat )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T2
              = ( some_nat @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o3780387683879180358Ti_nat @ F @ F3 @ T2 ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4199_TBOUND__option__case,axiom,
    ! [T2: option_num,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: num > heap_T8145700208782473153_VEBTi,Bnd2: num > nat] :
      ( ( ( T2 = none_num )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T2
              = ( some_num @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o337719470034958992Ti_num @ F @ F3 @ T2 ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4200_TBOUND__option__case,axiom,
    ! [T2: option_nat,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: nat > heap_T2636463487746394924on_nat,Bnd2: nat > nat] :
      ( ( ( T2 = none_nat )
       => ( time_T8353473612707095248on_nat @ F @ Bnd ) )
     => ( ! [X3: nat] :
            ( ( T2
              = ( some_nat @ X3 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T8353473612707095248on_nat @ ( case_o2256915875499652529at_nat @ F @ F3 @ T2 ) @ ( case_option_nat_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4201_TBOUND__option__case,axiom,
    ! [T2: option_num,F: heap_T2636463487746394924on_nat,Bnd: nat,F3: num > heap_T2636463487746394924on_nat,Bnd2: num > nat] :
      ( ( ( T2 = none_num )
       => ( time_T8353473612707095248on_nat @ F @ Bnd ) )
     => ( ! [X3: num] :
            ( ( T2
              = ( some_num @ X3 ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T8353473612707095248on_nat @ ( case_o8037619698510206971at_num @ F @ F3 @ T2 ) @ ( case_option_nat_num @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4202_TBOUND__option__case,axiom,
    ! [T2: option4927543243414619207at_nat,F: heap_Time_Heap_nat,Bnd: nat,F3: product_prod_nat_nat > heap_Time_Heap_nat,Bnd2: product_prod_nat_nat > nat] :
      ( ( ( T2 = none_P5556105721700978146at_nat )
       => ( time_TBOUND_nat @ F @ Bnd ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( T2
              = ( some_P7363390416028606310at_nat @ X3 ) )
           => ( time_TBOUND_nat @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_TBOUND_nat @ ( case_o3959993630158478256at_nat @ F @ F3 @ T2 ) @ ( case_o2098746482150326116at_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4203_TBOUND__option__case,axiom,
    ! [T2: option4927543243414619207at_nat,F: heap_T8145700208782473153_VEBTi,Bnd: nat,F3: product_prod_nat_nat > heap_T8145700208782473153_VEBTi,Bnd2: product_prod_nat_nat > nat] :
      ( ( ( T2 = none_P5556105721700978146at_nat )
       => ( time_T5737551269749752165_VEBTi @ F @ Bnd ) )
     => ( ! [X3: product_prod_nat_nat] :
            ( ( T2
              = ( some_P7363390416028606310at_nat @ X3 ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ X3 ) @ ( Bnd2 @ X3 ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( case_o1356590567247012107at_nat @ F @ F3 @ T2 ) @ ( case_o2098746482150326116at_nat @ Bnd @ Bnd2 @ T2 ) ) ) ) ).

% TBOUND_option_case
thf(fact_4204_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,T2: nat] :
      ( ( P
       => ( time_T8353473612707095248on_nat @ M @ T2 ) )
     => ( time_T8353473612707095248on_nat
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ ( if_nat @ P @ T2 @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4205_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,T2: nat] :
      ( ( P
       => ( time_TBOUND_nat @ M @ T2 ) )
     => ( time_TBOUND_nat
        @ ( heap_T3781436268274291734it_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ ( if_nat @ P @ T2 @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4206_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_Time_Heap_o,T2: nat] :
      ( ( P
       => ( time_TBOUND_o @ M @ T2 ) )
     => ( time_TBOUND_o
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ ( if_nat @ P @ T2 @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4207_TBOUND__assert_H__bind__strong,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,T2: nat] :
      ( ( P
       => ( time_T5737551269749752165_VEBTi @ M @ T2 ) )
     => ( time_T5737551269749752165_VEBTi
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ ( if_nat @ P @ T2 @ zero_zero_nat ) ) ) ).

% TBOUND_assert'_bind_strong
thf(fact_4208_pred__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% pred_bound_size_univ
thf(fact_4209_insert__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,N2: nat,U: real,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( U
          = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) ) @ ( plus_plus_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ U ) ) ) ) ) ) ) ).

% insert_bound_size_univ
thf(fact_4210_delete__correct,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_4211_delete__correct_H,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T2 @ X2 ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_4212_singleton__conv2,axiom,
    ! [A3: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ( ^ [Y5: vEBT_VEBT,Z2: vEBT_VEBT] : Y5 = Z2
          @ A3 ) )
      = ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv2
thf(fact_4213_singleton__conv2,axiom,
    ! [A3: complex] :
      ( ( collect_complex
        @ ( ^ [Y5: complex,Z2: complex] : Y5 = Z2
          @ A3 ) )
      = ( insert_complex @ A3 @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_4214_singleton__conv2,axiom,
    ! [A3: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ( ^ [Y5: product_prod_int_int,Z2: product_prod_int_int] : Y5 = Z2
          @ A3 ) )
      = ( insert5033312907999012233nt_int @ A3 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv2
thf(fact_4215_singleton__conv2,axiom,
    ! [A3: real] :
      ( ( collect_real
        @ ( ^ [Y5: real,Z2: real] : Y5 = Z2
          @ A3 ) )
      = ( insert_real @ A3 @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_4216_singleton__conv2,axiom,
    ! [A3: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z2: nat] : Y5 = Z2
          @ A3 ) )
      = ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_4217_singleton__conv2,axiom,
    ! [A3: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z2: int] : Y5 = Z2
          @ A3 ) )
      = ( insert_int @ A3 @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_4218_singleton__conv,axiom,
    ! [A3: vEBT_VEBT] :
      ( ( collect_VEBT_VEBT
        @ ^ [X: vEBT_VEBT] : X = A3 )
      = ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ).

% singleton_conv
thf(fact_4219_singleton__conv,axiom,
    ! [A3: complex] :
      ( ( collect_complex
        @ ^ [X: complex] : X = A3 )
      = ( insert_complex @ A3 @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_4220_singleton__conv,axiom,
    ! [A3: product_prod_int_int] :
      ( ( collec213857154873943460nt_int
        @ ^ [X: product_prod_int_int] : X = A3 )
      = ( insert5033312907999012233nt_int @ A3 @ bot_bo1796632182523588997nt_int ) ) ).

% singleton_conv
thf(fact_4221_singleton__conv,axiom,
    ! [A3: real] :
      ( ( collect_real
        @ ^ [X: real] : X = A3 )
      = ( insert_real @ A3 @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_4222_singleton__conv,axiom,
    ! [A3: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : X = A3 )
      = ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_4223_singleton__conv,axiom,
    ! [A3: int] :
      ( ( collect_int
        @ ^ [X: int] : X = A3 )
      = ( insert_int @ A3 @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_4224_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4225_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_4226_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4227_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_4228_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_4229_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_4230_ceiling__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_4231_ceiling__le__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_4232_ceiling__le__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_4233_ceiling__le__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_4234_zero__less__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).

% zero_less_ceiling
thf(fact_4235_zero__less__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% zero_less_ceiling
thf(fact_4236_numeral__less__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).

% numeral_less_ceiling
thf(fact_4237_numeral__less__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).

% numeral_less_ceiling
thf(fact_4238_ceiling__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_4239_ceiling__less__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
      = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_4240_one__le__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).

% one_le_ceiling
thf(fact_4241_one__le__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% one_le_ceiling
thf(fact_4242_ceiling__add__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_4243_ceiling__add__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_4244_ceiling__le__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
      = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_4245_ceiling__le__one,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
      = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_4246_one__less__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ one_one_rat @ X2 ) ) ).

% one_less_ceiling
thf(fact_4247_one__less__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ one_one_real @ X2 ) ) ).

% one_less_ceiling
thf(fact_4248_ceiling__add__one,axiom,
    ! [X2: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_4249_ceiling__add__one,axiom,
    ! [X2: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_4250_ceiling__diff__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_4251_ceiling__diff__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_4252_ceiling__numeral__power,axiom,
    ! [X2: num,N2: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ).

% ceiling_numeral_power
thf(fact_4253_ceiling__numeral__power,axiom,
    ! [X2: num,N2: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ).

% ceiling_numeral_power
thf(fact_4254_ceiling__diff__one,axiom,
    ! [X2: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_4255_ceiling__diff__one,axiom,
    ! [X2: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X2 @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_4256_ceiling__less__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_4257_ceiling__less__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_4258_numeral__le__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).

% numeral_le_ceiling
thf(fact_4259_numeral__le__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).

% numeral_le_ceiling
thf(fact_4260_insert__compr,axiom,
    ( insert_VEBT_VEBT
    = ( ^ [A2: vEBT_VEBT,B7: set_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [X: vEBT_VEBT] :
              ( ( X = A2 )
              | ( member_VEBT_VEBT @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4261_insert__compr,axiom,
    ( insert_real
    = ( ^ [A2: real,B7: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A2 )
              | ( member_real @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4262_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A2: nat,B7: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A2 )
              | ( member_nat @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4263_insert__compr,axiom,
    ( insert_int
    = ( ^ [A2: int,B7: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( X = A2 )
              | ( member_int @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4264_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A2: complex,B7: set_complex] :
          ( collect_complex
          @ ^ [X: complex] :
              ( ( X = A2 )
              | ( member_complex @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4265_insert__compr,axiom,
    ( insert5033312907999012233nt_int
    = ( ^ [A2: product_prod_int_int,B7: set_Pr958786334691620121nt_int] :
          ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( X = A2 )
              | ( member5262025264175285858nt_int @ X @ B7 ) ) ) ) ) ).

% insert_compr
thf(fact_4266_insert__Collect,axiom,
    ! [A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( insert_VEBT_VEBT @ A3 @ ( collect_VEBT_VEBT @ P ) )
      = ( collect_VEBT_VEBT
        @ ^ [U2: vEBT_VEBT] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4267_insert__Collect,axiom,
    ! [A3: real,P: real > $o] :
      ( ( insert_real @ A3 @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U2: real] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4268_insert__Collect,axiom,
    ! [A3: nat,P: nat > $o] :
      ( ( insert_nat @ A3 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4269_insert__Collect,axiom,
    ! [A3: int,P: int > $o] :
      ( ( insert_int @ A3 @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4270_insert__Collect,axiom,
    ! [A3: complex,P: complex > $o] :
      ( ( insert_complex @ A3 @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4271_insert__Collect,axiom,
    ! [A3: product_prod_int_int,P: product_prod_int_int > $o] :
      ( ( insert5033312907999012233nt_int @ A3 @ ( collec213857154873943460nt_int @ P ) )
      = ( collec213857154873943460nt_int
        @ ^ [U2: product_prod_int_int] :
            ( ( U2 != A3 )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_4272_Collect__conv__if2,axiom,
    ! [P: vEBT_VEBT > $o,A3: vEBT_VEBT] :
      ( ( ( P @ A3 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if2
thf(fact_4273_Collect__conv__if2,axiom,
    ! [P: complex > $o,A3: complex] :
      ( ( ( P @ A3 )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_complex @ A3 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_4274_Collect__conv__if2,axiom,
    ! [P: product_prod_int_int > $o,A3: product_prod_int_int] :
      ( ( ( P @ A3 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert5033312907999012233nt_int @ A3 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if2
thf(fact_4275_Collect__conv__if2,axiom,
    ! [P: real > $o,A3: real] :
      ( ( ( P @ A3 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_real @ A3 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_4276_Collect__conv__if2,axiom,
    ! [P: nat > $o,A3: nat] :
      ( ( ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_4277_Collect__conv__if2,axiom,
    ! [P: int > $o,A3: int] :
      ( ( ( P @ A3 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = ( insert_int @ A3 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A3 = X )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_4278_Collect__conv__if,axiom,
    ! [P: vEBT_VEBT > $o,A3: vEBT_VEBT] :
      ( ( ( P @ A3 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bo8194388402131092736T_VEBT ) ) ) ).

% Collect_conv_if
thf(fact_4279_Collect__conv__if,axiom,
    ! [P: complex > $o,A3: complex] :
      ( ( ( P @ A3 )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_complex @ A3 @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_complex
            @ ^ [X: complex] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_4280_Collect__conv__if,axiom,
    ! [P: product_prod_int_int > $o,A3: product_prod_int_int] :
      ( ( ( P @ A3 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert5033312907999012233nt_int @ A3 @ bot_bo1796632182523588997nt_int ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collec213857154873943460nt_int
            @ ^ [X: product_prod_int_int] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bo1796632182523588997nt_int ) ) ) ).

% Collect_conv_if
thf(fact_4281_Collect__conv__if,axiom,
    ! [P: real > $o,A3: real] :
      ( ( ( P @ A3 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_real @ A3 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_4282_Collect__conv__if,axiom,
    ! [P: nat > $o,A3: nat] :
      ( ( ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_4283_Collect__conv__if,axiom,
    ! [P: int > $o,A3: int] :
      ( ( ( P @ A3 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = ( insert_int @ A3 @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A3 )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A3 )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_4284_time__assert_H,axiom,
    ! [P: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_t4224138285095624986t_unit @ ( refine_Imp_assert @ P ) @ H2 )
      = zero_zero_nat ) ).

% time_assert'
thf(fact_4285_ceiling__mono,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ Y2 @ X2 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).

% ceiling_mono
thf(fact_4286_ceiling__mono,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ Y2 @ X2 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y2 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).

% ceiling_mono
thf(fact_4287_ceiling__less__cancel,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y2 ) )
     => ( ord_less_rat @ X2 @ Y2 ) ) ).

% ceiling_less_cancel
thf(fact_4288_ceiling__less__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y2 ) )
     => ( ord_less_real @ X2 @ Y2 ) ) ).

% ceiling_less_cancel
thf(fact_4289_ceiling__add__le,axiom,
    ! [X2: rat,Y2: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y2 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y2 ) ) ) ).

% ceiling_add_le
thf(fact_4290_ceiling__add__le,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y2 ) ) ) ).

% ceiling_add_le
thf(fact_4291_atLeast0__atMost__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_4292_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_4293_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_4294_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_4295_remove__subset,axiom,
    ! [X2: vEBT_VEBT,S4: set_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ S4 )
     => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ S4 ) ) ).

% remove_subset
thf(fact_4296_remove__subset,axiom,
    ! [X2: real,S4: set_real] :
      ( ( member_real @ X2 @ S4 )
     => ( ord_less_set_real @ ( minus_minus_set_real @ S4 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ S4 ) ) ).

% remove_subset
thf(fact_4297_remove__subset,axiom,
    ! [X2: int,S4: set_int] :
      ( ( member_int @ X2 @ S4 )
     => ( ord_less_set_int @ ( minus_minus_set_int @ S4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ S4 ) ) ).

% remove_subset
thf(fact_4298_remove__subset,axiom,
    ! [X2: nat,S4: set_nat] :
      ( ( member_nat @ X2 @ S4 )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ S4 ) ) ).

% remove_subset
thf(fact_4299_psubset__insert__iff,axiom,
    ! [A4: set_VEBT_VEBT,X2: vEBT_VEBT,B6: set_VEBT_VEBT] :
      ( ( ord_le3480810397992357184T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X2 @ B6 ) )
      = ( ( ( member_VEBT_VEBT @ X2 @ B6 )
         => ( ord_le3480810397992357184T_VEBT @ A4 @ B6 ) )
        & ( ~ ( member_VEBT_VEBT @ X2 @ B6 )
         => ( ( ( member_VEBT_VEBT @ X2 @ A4 )
             => ( ord_le3480810397992357184T_VEBT @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) @ B6 ) )
            & ( ~ ( member_VEBT_VEBT @ X2 @ A4 )
             => ( ord_le4337996190870823476T_VEBT @ A4 @ B6 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4300_psubset__insert__iff,axiom,
    ! [A4: set_real,X2: real,B6: set_real] :
      ( ( ord_less_set_real @ A4 @ ( insert_real @ X2 @ B6 ) )
      = ( ( ( member_real @ X2 @ B6 )
         => ( ord_less_set_real @ A4 @ B6 ) )
        & ( ~ ( member_real @ X2 @ B6 )
         => ( ( ( member_real @ X2 @ A4 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B6 ) )
            & ( ~ ( member_real @ X2 @ A4 )
             => ( ord_less_eq_set_real @ A4 @ B6 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4301_psubset__insert__iff,axiom,
    ! [A4: set_int,X2: int,B6: set_int] :
      ( ( ord_less_set_int @ A4 @ ( insert_int @ X2 @ B6 ) )
      = ( ( ( member_int @ X2 @ B6 )
         => ( ord_less_set_int @ A4 @ B6 ) )
        & ( ~ ( member_int @ X2 @ B6 )
         => ( ( ( member_int @ X2 @ A4 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B6 ) )
            & ( ~ ( member_int @ X2 @ A4 )
             => ( ord_less_eq_set_int @ A4 @ B6 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4302_psubset__insert__iff,axiom,
    ! [A4: set_nat,X2: nat,B6: set_nat] :
      ( ( ord_less_set_nat @ A4 @ ( insert_nat @ X2 @ B6 ) )
      = ( ( ( member_nat @ X2 @ B6 )
         => ( ord_less_set_nat @ A4 @ B6 ) )
        & ( ~ ( member_nat @ X2 @ B6 )
         => ( ( ( member_nat @ X2 @ A4 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B6 ) )
            & ( ~ ( member_nat @ X2 @ A4 )
             => ( ord_less_eq_set_nat @ A4 @ B6 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_4303_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info4 @ zero_zero_nat @ Ts @ S ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
thf(fact_4304_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv3: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ Uu2 @ Uv3 ) @ zero_zero_nat )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
thf(fact_4305_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv3: $o,Uw3: $o,N2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uv3 @ Uw3 ) @ ( suc @ N2 ) )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
thf(fact_4306_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
thf(fact_4307_mult__ceiling__le,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A3 @ B3 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A3 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_4308_mult__ceiling__le,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A3 @ B3 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A3 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_4309_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
thf(fact_4310_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn] :
      ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L2 ) )
     => ( ( insert5290817439147925377n_assn @ ( nth_Pr1769885009046257848n_assn @ L2 @ I ) @ ( set_Pr1139785259514867910n_assn @ ( list_u4534839942911652127n_assn @ L2 @ I @ X2 ) ) )
        = ( insert5290817439147925377n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4311_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
     => ( ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ L2 @ I ) @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X2 ) ) )
        = ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4312_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBTi,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s7982070591426661849_VEBTi @ L2 ) )
     => ( ( insert_VEBT_VEBTi @ ( nth_VEBT_VEBTi @ L2 @ I ) @ ( set_VEBT_VEBTi2 @ ( list_u6098035379799741383_VEBTi @ L2 @ I @ X2 ) ) )
        = ( insert_VEBT_VEBTi @ X2 @ ( set_VEBT_VEBTi2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4313_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_real,X2: real] :
      ( ( ord_less_nat @ I @ ( size_size_list_real @ L2 ) )
     => ( ( insert_real @ ( nth_real @ L2 @ I ) @ ( set_real2 @ ( list_update_real @ L2 @ I @ X2 ) ) )
        = ( insert_real @ X2 @ ( set_real2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4314_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_o,X2: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ L2 ) )
     => ( ( insert_o @ ( nth_o @ L2 @ I ) @ ( set_o2 @ ( list_update_o @ L2 @ I @ X2 ) ) )
        = ( insert_o @ X2 @ ( set_o2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4315_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_nat,X2: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
     => ( ( insert_nat @ ( nth_nat @ L2 @ I ) @ ( set_nat2 @ ( list_update_nat @ L2 @ I @ X2 ) ) )
        = ( insert_nat @ X2 @ ( set_nat2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4316_insert__swap__set__eq,axiom,
    ! [I: nat,L2: list_int,X2: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L2 ) )
     => ( ( insert_int @ ( nth_int @ L2 @ I ) @ ( set_int2 @ ( list_update_int @ L2 @ I @ X2 ) ) )
        = ( insert_int @ X2 @ ( set_int2 @ L2 ) ) ) ) ).

% insert_swap_set_eq
thf(fact_4317_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ Info4 @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X2 )
      = one_one_nat ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
thf(fact_4318_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,Va2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va2 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
thf(fact_4319_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Leaf @ A3 @ B3 ) @ X2 )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( X2 = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
thf(fact_4320_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
thf(fact_4321_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B3: $o] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
thf(fact_4322_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
thf(fact_4323_TBOUND__mono,axiom,
    ! [C: heap_T2636463487746394924on_nat,T2: nat,T4: nat] :
      ( ( time_T8353473612707095248on_nat @ C @ T2 )
     => ( ( ord_less_eq_nat @ T2 @ T4 )
       => ( time_T8353473612707095248on_nat @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_4324_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_nat,T2: nat,T4: nat] :
      ( ( time_TBOUND_nat @ C @ T2 )
     => ( ( ord_less_eq_nat @ T2 @ T4 )
       => ( time_TBOUND_nat @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_4325_TBOUND__mono,axiom,
    ! [C: heap_Time_Heap_o,T2: nat,T4: nat] :
      ( ( time_TBOUND_o @ C @ T2 )
     => ( ( ord_less_eq_nat @ T2 @ T4 )
       => ( time_TBOUND_o @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_4326_TBOUND__mono,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T2: nat,T4: nat] :
      ( ( time_T5737551269749752165_VEBTi @ C @ T2 )
     => ( ( ord_less_eq_nat @ T2 @ T4 )
       => ( time_T5737551269749752165_VEBTi @ C @ T4 ) ) ) ).

% TBOUND_mono
thf(fact_4327_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A3: $o,Uw3: $o] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Leaf @ A3 @ Uw3 ) @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
thf(fact_4328_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = one_one_nat ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
thf(fact_4329_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = one_one_nat ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
thf(fact_4330_TBOUND__refines,axiom,
    ! [C: heap_Time_Heap_nat,T2: nat,D: heap_Time_Heap_nat] :
      ( ( time_TBOUND_nat @ C @ T2 )
     => ( ( refine1365783493865988805es_nat @ C @ D )
       => ( time_TBOUND_nat @ D @ T2 ) ) ) ).

% TBOUND_refines
thf(fact_4331_TBOUND__refines,axiom,
    ! [C: heap_T2636463487746394924on_nat,T2: nat,D: heap_T2636463487746394924on_nat] :
      ( ( time_T8353473612707095248on_nat @ C @ T2 )
     => ( ( refine7594492741263601813on_nat @ C @ D )
       => ( time_T8353473612707095248on_nat @ D @ T2 ) ) ) ).

% TBOUND_refines
thf(fact_4332_TBOUND__refines,axiom,
    ! [C: heap_Time_Heap_o,T2: nat,D: heap_Time_Heap_o] :
      ( ( time_TBOUND_o @ C @ T2 )
     => ( ( refine_Imp_refines_o @ C @ D )
       => ( time_TBOUND_o @ D @ T2 ) ) ) ).

% TBOUND_refines
thf(fact_4333_TBOUND__refines,axiom,
    ! [C: heap_T8145700208782473153_VEBTi,T2: nat,D: heap_T8145700208782473153_VEBTi] :
      ( ( time_T5737551269749752165_VEBTi @ C @ T2 )
     => ( ( refine5565527176597971370_VEBTi @ C @ D )
       => ( time_T5737551269749752165_VEBTi @ D @ T2 ) ) ) ).

% TBOUND_refines
thf(fact_4334_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
thf(fact_4335_insersimp,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Y2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ Y2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insersimp
thf(fact_4336_insertsimp,axiom,
    ! [T2: vEBT_VEBT,N2: nat,L2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ( vEBT_VEBT_minNull @ T2 )
       => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ) ).

% insertsimp
thf(fact_4337_TBOUND__return,axiom,
    ! [X2: option_nat] : ( time_T8353473612707095248on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4338_TBOUND__return,axiom,
    ! [X2: nat] : ( time_TBOUND_nat @ ( heap_Time_return_nat @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4339_TBOUND__return,axiom,
    ! [X2: $o] : ( time_TBOUND_o @ ( heap_Time_return_o @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4340_TBOUND__return,axiom,
    ! [X2: vEBT_VEBTi] : ( time_T5737551269749752165_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ one_one_nat ) ).

% TBOUND_return
thf(fact_4341_time__return,axiom,
    ! [X2: product_unit,H2: heap_e7401611519738050253t_unit] :
      ( ( time_t4224138285095624986t_unit @ ( heap_T7507251653302230130t_unit @ X2 ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_4342_time__return,axiom,
    ! [X2: option_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_option_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_4343_time__return,axiom,
    ! [X2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_nat @ ( heap_Time_return_nat @ X2 ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_4344_time__return,axiom,
    ! [X2: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_o @ ( heap_Time_return_o @ X2 ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_4345_time__return,axiom,
    ! [X2: vEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_VEBT_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ H2 )
      = one_one_nat ) ).

% time_return
thf(fact_4346_TBOUND__nth,axiom,
    ! [Xs2: array_option_nat,I: nat] : ( time_T8353473612707095248on_nat @ ( array_nth_option_nat @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4347_TBOUND__nth,axiom,
    ! [Xs2: array_nat,I: nat] : ( time_TBOUND_nat @ ( array_nth_nat @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4348_TBOUND__nth,axiom,
    ! [Xs2: array_o,I: nat] : ( time_TBOUND_o @ ( array_nth_o @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4349_TBOUND__nth,axiom,
    ! [Xs2: array_VEBT_VEBTi,I: nat] : ( time_T5737551269749752165_VEBTi @ ( array_nth_VEBT_VEBTi @ Xs2 @ I ) @ one_one_nat ) ).

% TBOUND_nth
thf(fact_4350_TBOUND__def,axiom,
    ( time_T7469515765551943773t_unit
    = ( ^ [M3: heap_T5738788834812785303t_unit,T: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M3 @ H ) @ T ) ) ) ).

% TBOUND_def
thf(fact_4351_TBOUND__def,axiom,
    ( time_T8353473612707095248on_nat
    = ( ^ [M3: heap_T2636463487746394924on_nat,T: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M3 @ H ) @ T ) ) ) ).

% TBOUND_def
thf(fact_4352_TBOUND__def,axiom,
    ( time_TBOUND_nat
    = ( ^ [M3: heap_Time_Heap_nat,T: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M3 @ H ) @ T ) ) ) ).

% TBOUND_def
thf(fact_4353_TBOUND__def,axiom,
    ( time_TBOUND_o
    = ( ^ [M3: heap_Time_Heap_o,T: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M3 @ H ) @ T ) ) ) ).

% TBOUND_def
thf(fact_4354_TBOUND__def,axiom,
    ( time_T5737551269749752165_VEBTi
    = ( ^ [M3: heap_T8145700208782473153_VEBTi,T: nat] :
        ! [H: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M3 @ H ) @ T ) ) ) ).

% TBOUND_def
thf(fact_4355_TBOUNDI,axiom,
    ! [M: heap_T5738788834812785303t_unit,T2: nat] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M @ H6 ) @ T2 )
     => ( time_T7469515765551943773t_unit @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_4356_TBOUNDI,axiom,
    ! [M: heap_T2636463487746394924on_nat,T2: nat] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H6 ) @ T2 )
     => ( time_T8353473612707095248on_nat @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_4357_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_nat,T2: nat] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_nat @ M @ H6 ) @ T2 )
     => ( time_TBOUND_nat @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_4358_TBOUNDI,axiom,
    ! [M: heap_Time_Heap_o,T2: nat] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_o @ M @ H6 ) @ T2 )
     => ( time_TBOUND_o @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_4359_TBOUNDI,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T2: nat] :
      ( ! [H6: heap_e7401611519738050253t_unit] : ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H6 ) @ T2 )
     => ( time_T5737551269749752165_VEBTi @ M @ T2 ) ) ).

% TBOUNDI
thf(fact_4360_TBOUNDD,axiom,
    ! [M: heap_T5738788834812785303t_unit,T2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T7469515765551943773t_unit @ M @ T2 )
     => ( ord_less_eq_nat @ ( time_t4224138285095624986t_unit @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_4361_TBOUNDD,axiom,
    ! [M: heap_T2636463487746394924on_nat,T2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T8353473612707095248on_nat @ M @ T2 )
     => ( ord_less_eq_nat @ ( time_time_option_nat @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_4362_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_nat,T2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_nat @ M @ T2 )
     => ( ord_less_eq_nat @ ( time_time_nat @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_4363_TBOUNDD,axiom,
    ! [M: heap_Time_Heap_o,T2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_TBOUND_o @ M @ T2 )
     => ( ord_less_eq_nat @ ( time_time_o @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_4364_TBOUNDD,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,T2: nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_T5737551269749752165_VEBTi @ M @ T2 )
     => ( ord_less_eq_nat @ ( time_time_VEBT_VEBTi @ M @ H2 ) @ T2 ) ) ).

% TBOUNDD
thf(fact_4365_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_4366_ceiling__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).

% ceiling_log2_div2
thf(fact_4367_TBOUND__prod__case,axiom,
    ! [T2: product_prod_num_num,F: num > num > heap_Time_Heap_nat,Bnd: num > num > nat] :
      ( ! [A: num,B: num] :
          ( ( T2
            = ( product_Pair_num_num @ A @ B ) )
         => ( time_TBOUND_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_nat @ ( produc3422757380792381434ap_nat @ F @ T2 ) @ ( produc2914010905598588082um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4368_TBOUND__prod__case,axiom,
    ! [T2: product_prod_nat_num,F: nat > num > heap_Time_Heap_nat,Bnd: nat > num > nat] :
      ( ! [A: nat,B: num] :
          ( ( T2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( time_TBOUND_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_nat @ ( produc8139307783169301188ap_nat @ F @ T2 ) @ ( produc4973203077627929192um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4369_TBOUND__prod__case,axiom,
    ! [T2: product_prod_nat_nat,F: nat > nat > heap_Time_Heap_nat,Bnd: nat > nat > nat] :
      ( ! [A: nat,B: nat] :
          ( ( T2
            = ( product_Pair_nat_nat @ A @ B ) )
         => ( time_TBOUND_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_nat @ ( produc3495815207102967182ap_nat @ F @ T2 ) @ ( produc6842872674320459806at_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4370_TBOUND__prod__case,axiom,
    ! [T2: product_prod_int_int,F: int > int > heap_Time_Heap_nat,Bnd: int > int > nat] :
      ( ! [A: int,B: int] :
          ( ( T2
            = ( product_Pair_int_int @ A @ B ) )
         => ( time_TBOUND_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_nat @ ( produc2561121167688006358ap_nat @ F @ T2 ) @ ( produc8213879946458358998nt_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4371_TBOUND__prod__case,axiom,
    ! [T2: product_prod_num_num,F: num > num > heap_Time_Heap_o,Bnd: num > num > nat] :
      ( ! [A: num,B: num] :
          ( ( T2
            = ( product_Pair_num_num @ A @ B ) )
         => ( time_TBOUND_o @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_o @ ( produc2098883955491630340Heap_o @ F @ T2 ) @ ( produc2914010905598588082um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4372_TBOUND__prod__case,axiom,
    ! [T2: product_prod_nat_num,F: nat > num > heap_Time_Heap_o,Bnd: nat > num > nat] :
      ( ! [A: nat,B: num] :
          ( ( T2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( time_TBOUND_o @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_o @ ( produc1066491148394503098Heap_o @ F @ T2 ) @ ( produc4973203077627929192um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4373_TBOUND__prod__case,axiom,
    ! [T2: product_prod_nat_nat,F: nat > nat > heap_Time_Heap_o,Bnd: nat > nat > nat] :
      ( ! [A: nat,B: nat] :
          ( ( T2
            = ( product_Pair_nat_nat @ A @ B ) )
         => ( time_TBOUND_o @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_o @ ( produc3505292621261808240Heap_o @ F @ T2 ) @ ( produc6842872674320459806at_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4374_TBOUND__prod__case,axiom,
    ! [T2: product_prod_int_int,F: int > int > heap_Time_Heap_o,Bnd: int > int > nat] :
      ( ! [A: int,B: int] :
          ( ( T2
            = ( product_Pair_int_int @ A @ B ) )
         => ( time_TBOUND_o @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_TBOUND_o @ ( produc2625828211853636392Heap_o @ F @ T2 ) @ ( produc8213879946458358998nt_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4375_TBOUND__prod__case,axiom,
    ! [T2: product_prod_num_num,F: num > num > heap_T8145700208782473153_VEBTi,Bnd: num > num > nat] :
      ( ! [A: num,B: num] :
          ( ( T2
            = ( product_Pair_num_num @ A @ B ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_T5737551269749752165_VEBTi @ ( produc1633297657014491743_VEBTi @ F @ T2 ) @ ( produc2914010905598588082um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4376_TBOUND__prod__case,axiom,
    ! [T2: product_prod_nat_num,F: nat > num > heap_T8145700208782473153_VEBTi,Bnd: nat > num > nat] :
      ( ! [A: nat,B: num] :
          ( ( T2
            = ( product_Pair_nat_num @ A @ B ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( time_T5737551269749752165_VEBTi @ ( produc8100297318048944937_VEBTi @ F @ T2 ) @ ( produc4973203077627929192um_nat @ Bnd @ T2 ) ) ) ).

% TBOUND_prod_case
thf(fact_4377_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_4378_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T2636463487746394924on_nat,T2: nat] :
      ( ( P
       => ( time_T8353473612707095248on_nat @ M @ T2 ) )
     => ( time_T8353473612707095248on_nat
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ T2 ) ) ).

% TBOUND_assert'_weak
thf(fact_4379_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_nat,T2: nat] :
      ( ( P
       => ( time_TBOUND_nat @ M @ T2 ) )
     => ( time_TBOUND_nat
        @ ( heap_T3781436268274291734it_nat @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ T2 ) ) ).

% TBOUND_assert'_weak
thf(fact_4380_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_Time_Heap_o,T2: nat] :
      ( ( P
       => ( time_TBOUND_o @ M @ T2 ) )
     => ( time_TBOUND_o
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ T2 ) ) ).

% TBOUND_assert'_weak
thf(fact_4381_TBOUND__assert_H__weak,axiom,
    ! [P: $o,M: heap_T8145700208782473153_VEBTi,T2: nat] :
      ( ( P
       => ( time_T5737551269749752165_VEBTi @ M @ T2 ) )
     => ( time_T5737551269749752165_VEBTi
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ P )
          @ ^ [Uu: product_unit] : M )
        @ T2 ) ) ).

% TBOUND_assert'_weak
thf(fact_4382_pred__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_p_r_e_d @ T2 @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% pred_bound_height
thf(fact_4383_insert__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_i_n_s_e_r_t @ T2 @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% insert_bound_height
thf(fact_4384_succ__bound__height,axiom,
    ! [T2: vEBT_VEBT,N2: nat,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_eq_nat @ ( vEBT_T_s_u_c_c @ T2 @ X2 ) @ ( times_times_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_height @ T2 ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% succ_bound_height
thf(fact_4385_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2
           != ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
               => ( Y2
                 != ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                   => ( Y2
                     != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                        @ ( if_nat
                          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                            & ~ ( ( Xa = Mi2 )
                                | ( Xa = Ma2 ) ) )
                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                          @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
thf(fact_4386_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_i_n_s_e_r_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ~ ( ( X2 = Mi )
                | ( X2 = Ma ) ) )
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
          @ one_one_nat ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
thf(fact_4387_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_i_n_s_e_r_t @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ ( Xa = zero_zero_nat ) @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
             => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) )
                   => ( ( Y2
                        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( ( Y2
                          = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                & ~ ( ( Xa = Mi2 )
                                    | ( Xa = Ma2 ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_i_n_s_e_r_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_T_m_i_n_N_u_l_l @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( if_nat @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_T_i_n_s_e_r_t @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ one_one_nat ) )
                              @ one_one_nat ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T9217963907923527482_t_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
thf(fact_4388_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ B ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2
             != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
           => ( ? [N4: nat] :
                  ( Xa
                  = ( suc @ N4 ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y2 != one_one_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                     => ( Y2
                       != ( plus_plus_nat @ one_one_nat
                          @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                              @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ one_one_nat
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
thf(fact_4389_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A: $o,Uw2: $o] :
                ( X2
                = ( vEBT_Leaf @ A @ Uw2 ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2
               != ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ? [Va3: nat] :
                      ( Xa
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ( Y2
                   != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ) )
           => ( ( ? [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ( Y2
                         != ( plus_plus_nat @ one_one_nat
                            @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                              @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                    @ ( if_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                        @ ( if_nat
                                          @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                            = none_nat )
                                          @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                  @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
thf(fact_4390_log__ceil__idem,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
        = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ) ) ).

% log_ceil_idem
thf(fact_4391_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_s_u_c_c @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                  @ ( if_nat
                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                       != none_nat )
                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                      @ ( if_nat
                        @ ( ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          = none_nat )
                        @ one_one_nat
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
thf(fact_4392_highsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_high @ X2 @ N2 ) )
      = ( vEBT_VEBT_highi @ X2 @ N2 ) ) ).

% highsimp
thf(fact_4393_lowsimp,axiom,
    ! [X2: nat,N2: nat] :
      ( ( heap_Time_return_nat @ ( vEBT_VEBT_low @ X2 @ N2 ) )
      = ( vEBT_VEBT_lowi @ X2 @ N2 ) ) ).

% lowsimp
thf(fact_4394_highi__def,axiom,
    ( vEBT_VEBT_highi
    = ( ^ [X: nat,N: nat] : ( heap_Time_return_nat @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% highi_def
thf(fact_4395_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_4396_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_4397_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_4398_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_4399_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_4400_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_4401_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_4402_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_4403_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_4404_of__int__0,axiom,
    ( ( ring_17408606157368542149l_num1 @ zero_zero_int )
    = zero_z3563351764282998399l_num1 ) ).

% of_int_0
thf(fact_4405_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4406_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4407_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_4408_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ K ) )
      = ( numera7442385471795722001l_num1 @ K ) ) ).

% of_int_numeral
thf(fact_4409_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_4410_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_4411_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_4412_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_4413_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_4414_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_4415_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_4416_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_4417_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_4418_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_4419_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_4420_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_4421_of__int__1,axiom,
    ( ( ring_1_of_int_uint32 @ one_one_int )
    = one_one_uint32 ) ).

% of_int_1
thf(fact_4422_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_4423_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_4424_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_4425_of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% of_int_1
thf(fact_4426_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X2 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W ) )
      = ( X2
        = ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4427_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X2 )
        = ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) )
      = ( X2
        = ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4428_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X2 )
        = ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W ) )
      = ( X2
        = ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4429_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X2 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B3 ) @ W ) )
      = ( X2
        = ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4430_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ( ring_18347121197199848620nteger @ X2 )
        = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W ) )
      = ( X2
        = ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_4431_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W )
        = ( ring_1_of_int_rat @ X2 ) )
      = ( ( power_power_int @ B3 @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4432_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W )
        = ( ring_1_of_int_int @ X2 ) )
      = ( ( power_power_int @ B3 @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4433_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W )
        = ( ring_1_of_int_real @ X2 ) )
      = ( ( power_power_int @ B3 @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4434_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B3 ) @ W )
        = ( ring_17405671764205052669omplex @ X2 ) )
      = ( ( power_power_int @ B3 @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4435_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W )
        = ( ring_18347121197199848620nteger @ X2 ) )
      = ( ( power_power_int @ B3 @ W )
        = X2 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_4436_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4437_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_17408606157368542149l_num1 @ ( power_power_int @ Z @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( ring_17408606157368542149l_num1 @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4438_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4439_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4440_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4441_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_18347121197199848620nteger @ ( power_power_int @ Z @ N2 ) )
      = ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_4442_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_4443_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_4444_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_4445_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_4446_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_4447_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_4448_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4449_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4450_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_4451_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4452_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4453_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_4454_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_4455_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_4456_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_4457_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_4458_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_4459_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_4460_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4461_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4462_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_4463_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4464_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4465_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_4466_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_4467_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_4468_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_4469_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_4470_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_4471_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_4472_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_4473_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_4474_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_4475_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_4476_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_4477_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_4478_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y2 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4479_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y2 )
        = ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4480_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y2 )
        = ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4481_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y2 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4482_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y2 )
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_4483_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X2 ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4484_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4485_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 )
        = ( ring_1_of_int_real @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4486_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 )
        = ( ring_1_of_int_rat @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4487_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = ( ring_1_of_int_int @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_4488_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4489_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4490_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4491_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) )
      = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_4492_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4493_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W ) @ ( ring_18347121197199848620nteger @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4494_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4495_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_4496_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4497_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4498_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4499_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X2: int,B3: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) )
      = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_4500_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( ring_18347121197199848620nteger @ B3 ) @ W ) @ ( ring_18347121197199848620nteger @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4501_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4502_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4503_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B3: int,W: nat,X2: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( power_power_int @ B3 @ W ) @ X2 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_4504_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4505_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4506_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4507_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_4508_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) @ ( ring_18347121197199848620nteger @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4509_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4510_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4511_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_4512_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4513_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4514_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4515_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_4516_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N2 ) @ ( ring_18347121197199848620nteger @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4517_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N2 ) @ ( ring_1_of_int_real @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4518_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N2 ) @ ( ring_1_of_int_rat @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4519_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ ( ring_1_of_int_int @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_4520_ex__le__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_le_of_int
thf(fact_4521_ex__le__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_le_of_int
thf(fact_4522_ex__of__int__less,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_4523_ex__of__int__less,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 ) ).

% ex_of_int_less
thf(fact_4524_ex__less__of__int,axiom,
    ! [X2: real] :
    ? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).

% ex_less_of_int
thf(fact_4525_ex__less__of__int,axiom,
    ! [X2: rat] :
    ? [Z4: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).

% ex_less_of_int
thf(fact_4526_le__of__int__ceiling,axiom,
    ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).

% le_of_int_ceiling
thf(fact_4527_le__of__int__ceiling,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).

% le_of_int_ceiling
thf(fact_4528_ceiling__le__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
      = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_4529_ceiling__le__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
      = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_4530_ceiling__le,axiom,
    ! [X2: real,A3: int] :
      ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A3 ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A3 ) ) ).

% ceiling_le
thf(fact_4531_ceiling__le,axiom,
    ! [X2: rat,A3: int] :
      ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A3 ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A3 ) ) ).

% ceiling_le
thf(fact_4532_less__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).

% less_ceiling_iff
thf(fact_4533_less__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).

% less_ceiling_iff
thf(fact_4534_real__of__int__div4,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) ) ) ).

% real_of_int_div4
thf(fact_4535_minNull__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_N_u_l_l @ T2 ) @ one_one_nat ) ).

% minNull_bound
thf(fact_4536_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $false @ $false ) )
    = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
thf(fact_4537_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv3: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ $true @ Uv3 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
thf(fact_4538_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu2: $o] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Leaf @ Uu2 @ $true ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
thf(fact_4539_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_4540_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_4541_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_4542_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_4543_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_4544_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_4545_floor__exists1,axiom,
    ! [X2: real] :
    ? [X3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X2 )
            & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_4546_floor__exists1,axiom,
    ! [X2: rat] :
    ? [X3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X2 )
            & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
thf(fact_4547_floor__exists,axiom,
    ! [X2: real] :
    ? [Z4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
      & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4548_floor__exists,axiom,
    ! [X2: rat] :
    ? [Z4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
      & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_4549_of__int__ceiling__le__add__one,axiom,
    ! [R: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ ( plus_plus_real @ R @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4550_of__int__ceiling__le__add__one,axiom,
    ! [R: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ ( plus_plus_rat @ R @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_4551_of__int__ceiling__diff__one__le,axiom,
    ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R ) ) @ one_one_real ) @ R ) ).

% of_int_ceiling_diff_one_le
thf(fact_4552_of__int__ceiling__diff__one__le,axiom,
    ! [R: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R ) ) @ one_one_rat ) @ R ) ).

% of_int_ceiling_diff_one_le
thf(fact_4553_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_4554_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_4555_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_4556_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X2: int] :
      ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( ring_18347121197199848620nteger @ X2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X2 ) ) ).

% of_nat_less_of_int_iff
thf(fact_4557_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_4558_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).

% int_less_real_le
thf(fact_4559_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_4560_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Leaf @ A3 @ B3 ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B3 @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
thf(fact_4561_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_4562_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
thf(fact_4563_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: list_VEBT_VEBT,Uw3: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
thf(fact_4564_ceiling__correct,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
      & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).

% ceiling_correct
thf(fact_4565_ceiling__correct,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
      & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).

% ceiling_correct
thf(fact_4566_ceiling__unique,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X2 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_4567_ceiling__unique,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X2 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_4568_ceiling__eq__iff,axiom,
    ! [X2: real,A3: int] :
      ( ( ( archim7802044766580827645g_real @ X2 )
        = A3 )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A3 ) @ one_one_real ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A3 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_4569_ceiling__eq__iff,axiom,
    ! [X2: rat,A3: int] :
      ( ( ( archim2889992004027027881ng_rat @ X2 )
        = A3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A3 ) @ one_one_rat ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A3 ) ) ) ) ).

% ceiling_eq_iff
thf(fact_4570_ceiling__split,axiom,
    ! [P: int > $o,T2: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T2 ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T2 )
              & ( ord_less_eq_real @ T2 @ ( ring_1_of_int_real @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_4571_ceiling__split,axiom,
    ! [P: int > $o,T2: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T2 ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T2 )
              & ( ord_less_eq_rat @ T2 @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_4572_ceiling__less__iff,axiom,
    ! [X2: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_4573_ceiling__less__iff,axiom,
    ! [X2: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_4574_le__ceiling__iff,axiom,
    ! [Z: int,X2: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).

% le_ceiling_iff
thf(fact_4575_le__ceiling__iff,axiom,
    ! [Z: int,X2: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).

% le_ceiling_iff
thf(fact_4576_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb @ Vc ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
thf(fact_4577_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw3: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw3 @ Ux2 @ Uy2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
thf(fact_4578_real__of__int__div2,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) ) ) ).

% real_of_int_div2
thf(fact_4579_real__of__int__div3,axiom,
    ! [N2: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X2 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_4580_maxt__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_a_x_t @ T2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% maxt_bound
thf(fact_4581_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_4582_ceiling__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ Q2 ) ) ) ).

% ceiling_divide_upper
thf(fact_4583_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_a_x_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
thf(fact_4584_mint__bound,axiom,
    ! [T2: vEBT_VEBT] : ( ord_less_eq_nat @ ( vEBT_T_m_i_n_t @ T2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ).

% mint_bound
thf(fact_4585_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Leaf @ A3 @ B3 ) )
      = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A3 @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
thf(fact_4586_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
      ( ( vEBT_T_m_i_n_t @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = one_one_nat ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
thf(fact_4587_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q2 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q2 ) ) ) @ one_one_real ) @ Q2 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_4588_ceiling__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q2 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q2 ) ) ) @ one_one_rat ) @ Q2 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_4589_ceiling__eq,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X2 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4590_ceiling__eq,axiom,
    ! [N2: int,X2: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X2 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_4591_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leaf @ $false @ $false ) )
         => ( Y2 != one_one_nat ) )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => ( Y2 != one_one_nat ) )
         => ( ( ? [Uu3: $o] :
                  ( X2
                  = ( vEBT_Leaf @ Uu3 @ $true ) )
             => ( Y2 != one_one_nat ) )
           => ( ( ? [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
               => ( Y2 != one_one_nat ) )
             => ~ ( ? [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                 => ( Y2 != one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
thf(fact_4592_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X2 )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ( Y2
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
             => ( Y2 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
thf(fact_4593_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X2 )
        = Y2 )
     => ( ! [A: $o] :
            ( ? [B: $o] :
                ( X2
                = ( vEBT_Leaf @ A @ B ) )
           => ( Y2
             != ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X2
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2 != one_one_nat ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
             => ( Y2 != one_one_nat ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
thf(fact_4594_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_d_e_l_e_t_e @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
        @ ( if_nat
          @ ( ( ord_less_nat @ X2 @ Mi )
            | ( ord_less_nat @ Ma @ X2 ) )
          @ one_one_nat
          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
            @ ( if_nat
              @ ( ( X2 = Mi )
                & ( X2 = Ma ) )
              @ ( numeral_numeral_nat @ ( bit1 @ one ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary4 ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                            @ ( if_nat
                              @ ( ( ( X2 = Mi )
                                 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                                    = Ma ) )
                                & ( ( X2 != Mi )
                                 => ( X2 = Ma ) ) )
                              @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ( plus_plus_nat @ one_one_nat
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      = none_nat )
                                    @ one_one_nat
                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary4 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ one_one_nat ) ) )
                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                          @ ( if_nat
                            @ ( ( ( X2 = Mi )
                               => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) )
                                  = Ma ) )
                              & ( ( X2 != Mi )
                               => ( X2 = Ma ) ) )
                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary4 ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                            @ one_one_nat ) ) ) ) )
                  @ one_one_nat ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
thf(fact_4595_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( ( Xa = zero_zero_nat )
           => ( Y2 != one_one_nat ) ) )
       => ( ( ? [A: $o,B: $o] :
                ( X2
                = ( vEBT_Leaf @ A @ B ) )
           => ( ( Xa
                = ( suc @ zero_zero_nat ) )
             => ( Y2 != one_one_nat ) ) )
         => ( ( ? [A: $o,B: $o] :
                  ( X2
                  = ( vEBT_Leaf @ A @ B ) )
             => ( ? [N4: nat] :
                    ( Xa
                    = ( suc @ ( suc @ N4 ) ) )
               => ( Y2 != one_one_nat ) ) )
           => ( ( ? [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
               => ( Y2 != one_one_nat ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) )
                 => ( Y2 != one_one_nat ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) )
                   => ( Y2 != one_one_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ( Y2
                         != ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                            @ ( if_nat
                              @ ( ( ord_less_nat @ Xa @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa ) )
                              @ one_one_nat
                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( Xa = Mi2 )
                                    & ( Xa = Ma2 ) )
                                  @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                          @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                @ ( if_nat
                                                  @ ( ( ( Xa = Mi2 )
                                                     => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa != Mi2 )
                                                     => ( Xa = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ ( plus_plus_nat @ one_one_nat
                                                      @ ( if_nat
                                                        @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          = none_nat )
                                                        @ one_one_nat
                                                        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ one_one_nat ) ) )
                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                              @ ( if_nat
                                                @ ( ( ( Xa = Mi2 )
                                                   => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa != Mi2 )
                                                   => ( Xa = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                @ one_one_nat ) ) ) ) )
                                      @ one_one_nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
thf(fact_4596_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,X2: nat] :
      ( ( vEBT_T_p_r_e_d @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary4 ) @ X2 )
      = ( plus_plus_nat @ one_one_nat
        @ ( if_nat @ ( ord_less_nat @ Ma @ X2 ) @ one_one_nat
          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
            @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
              @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                @ ( if_nat
                  @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                     != none_nat )
                    & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                    @ ( if_nat
                      @ ( ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        = none_nat )
                      @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary4 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
              @ one_one_nat ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
thf(fact_4597_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_d_e_l_e_t_e @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,B: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ B ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [N4: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ N4 ) ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
             => ( ! [Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TreeList3 @ Summary ) @ Xa ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) )
                       => ( ( Y2 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ TreeList3 @ Summary ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( Y2
                              = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                @ ( if_nat
                                  @ ( ( ord_less_nat @ Xa @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa ) )
                                  @ one_one_nat
                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                    @ ( if_nat
                                      @ ( ( Xa = Mi2 )
                                        & ( Xa = Ma2 ) )
                                      @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( if_nat @ ( Xa = Mi2 ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( vEBT_T_m_i_n_t @ Summary ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) ) @ one_one_nat ) ) @ one_one_nat )
                                        @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_d_e_l_e_t_e @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_i_n_N_u_l_l @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                              @ ( if_nat @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_d_e_l_e_t_e @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                    @ ( if_nat
                                                      @ ( ( ( Xa = Mi2 )
                                                         => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                            = Ma2 ) )
                                                        & ( ( Xa != Mi2 )
                                                         => ( Xa = Ma2 ) ) )
                                                      @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                        @ ( plus_plus_nat @ one_one_nat
                                                          @ ( if_nat
                                                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              = none_nat )
                                                            @ one_one_nat
                                                            @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                                      @ one_one_nat ) ) )
                                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
                                                  @ ( if_nat
                                                    @ ( ( ( Xa = Mi2 )
                                                       => ( ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                                          = Ma2 ) )
                                                      & ( ( Xa != Mi2 )
                                                       => ( Xa = Ma2 ) ) )
                                                    @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa = Mi2 ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                    @ one_one_nat ) ) ) ) )
                                          @ one_one_nat ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T8441311223069195367_e_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
thf(fact_4598_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_s_u_c_c @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ B ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2
                    = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ B ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
               => ! [N4: nat] :
                    ( ( Xa
                      = ( suc @ N4 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
           => ( ! [Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux3 @ Uy3 @ Uz3 ) @ Xa ) ) ) )
             => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                       => ( ( Y2
                            = ( plus_plus_nat @ one_one_nat
                              @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ one_one_nat
                                @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) )
                                  @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                    @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_s_u_c_c @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_s_u_c_c @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ one_one_nat
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                    @ one_one_nat ) ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_s_u_c_c_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
thf(fact_4599_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: nat] :
      ( ( ( vEBT_T_p_r_e_d @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( Xa = zero_zero_nat )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A: $o,Uw2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ A @ Uw2 ) )
               => ( ( Xa
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y2
                      = ( plus_plus_nat @ one_one_nat @ one_one_nat ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A: $o,B: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ A @ B ) )
                 => ! [Va3: nat] :
                      ( ( Xa
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( Y2
                          = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy3: nat,Uz3: list_VEBT_VEBT,Va4: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy3 @ Uz3 @ Va4 ) @ Xa ) ) ) )
               => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa ) ) ) )
                 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X2
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y2 = one_one_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                          ( ( X2
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) )
                         => ( ( Y2
                              = ( plus_plus_nat @ one_one_nat
                                @ ( if_nat @ ( ord_less_nat @ Ma2 @ Xa ) @ one_one_nat
                                  @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ one_one_nat )
                                    @ ( if_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                                      @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( vEBT_T_m_i_n_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
                                        @ ( if_nat
                                          @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                             != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_p_r_e_d @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_T_p_r_e_d @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_nat )
                                            @ ( if_nat
                                              @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                = none_nat )
                                              @ ( plus_plus_nat @ one_one_nat @ one_one_nat )
                                              @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( vEBT_T_m_a_x_t @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                                      @ one_one_nat ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_T_p_r_e_d_rel2 @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
thf(fact_4600_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R6: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R6 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4601_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R6: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R6 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_4602_lemma__termdiff3,axiom,
    ! [H2: real,Z: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4603_lemma__termdiff3,axiom,
    ! [H2: complex,Z: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_4604_foldr__zero,axiom,
    ! [Xs2: list_nat,D: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( nth_nat @ Xs2 @ I2 ) ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( minus_minus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ D ) @ D ) ) ) ).

% foldr_zero
thf(fact_4605_foldr__one,axiom,
    ! [D: nat,Ys: list_nat] : ( ord_less_eq_nat @ D @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ).

% foldr_one
thf(fact_4606_foldr__same__int,axiom,
    ! [Xs2: list_nat,Y2: nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( X3 = Y2 ) )
       => ( ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ zero_zero_nat )
          = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Y2 ) ) ) ) ).

% foldr_same_int
thf(fact_4607_foldr__mono,axiom,
    ! [Xs2: list_nat,Ys: list_nat,C: nat,D: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
           => ( ord_less_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
       => ( ( ord_less_eq_nat @ C @ D )
         => ( ord_less_eq_nat @ ( plus_plus_nat @ ( foldr_nat_nat @ plus_plus_nat @ Xs2 @ C ) @ ( size_size_list_nat @ Ys ) ) @ ( foldr_nat_nat @ plus_plus_nat @ Ys @ D ) ) ) ) ) ).

% foldr_mono
thf(fact_4608_word__of__int__numeral,axiom,
    ! [Bin: num] :
      ( ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ Bin ) )
      = ( numera7442385471795722001l_num1 @ Bin ) ) ).

% word_of_int_numeral
thf(fact_4609_word__of__int__0,axiom,
    ( ( ring_17408606157368542149l_num1 @ zero_zero_int )
    = zero_z3563351764282998399l_num1 ) ).

% word_of_int_0
thf(fact_4610_word__of__int__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ one_one_int )
    = one_on7727431528512463931l_num1 ) ).

% word_of_int_1
thf(fact_4611_numeral__div__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ K @ L2 ) ) ) ).

% numeral_div_numeral
thf(fact_4612_numeral__div__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ K @ L2 ) ) ) ).

% numeral_div_numeral
thf(fact_4613_foldr__length,axiom,
    ! [L2: list_real] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_real @ L2 ) ) ).

% foldr_length
thf(fact_4614_foldr__length,axiom,
    ! [L2: list_o] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_o @ L2 ) ) ).

% foldr_length
thf(fact_4615_foldr__length,axiom,
    ! [L2: list_nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_nat @ L2 ) ) ).

% foldr_length
thf(fact_4616_foldr__length,axiom,
    ! [L2: list_int] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L2
        @ zero_zero_nat )
      = ( size_size_list_int @ L2 ) ) ).

% foldr_length
thf(fact_4617_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_4618_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_4619_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4620_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_4621_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4622_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
      = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_4623_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_4624_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_4625_one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( product_fst_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ).

% one_div_numeral
thf(fact_4626_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R6: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R6 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_4627_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R6: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R6 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_4628_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4629_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4630_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_4631_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4632_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4633_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_4634_word__numeral__alt,axiom,
    ( numera7442385471795722001l_num1
    = ( ^ [B2: num] : ( ring_17408606157368542149l_num1 @ ( numeral_numeral_int @ B2 ) ) ) ) ).

% word_numeral_alt
thf(fact_4635_word__of__int__power__hom,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_2184487114949457152l_num1 @ ( ring_17408606157368542149l_num1 @ A3 ) @ N2 )
      = ( ring_17408606157368542149l_num1 @ ( power_power_int @ A3 @ N2 ) ) ) ).

% word_of_int_power_hom
thf(fact_4636_foldr__length__aux,axiom,
    ! [L2: list_real,A3: nat] :
      ( ( foldr_real_nat
        @ ^ [X: real] : suc
        @ L2
        @ A3 )
      = ( plus_plus_nat @ A3 @ ( size_size_list_real @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4637_foldr__length__aux,axiom,
    ! [L2: list_o,A3: nat] :
      ( ( foldr_o_nat
        @ ^ [X: $o] : suc
        @ L2
        @ A3 )
      = ( plus_plus_nat @ A3 @ ( size_size_list_o @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4638_foldr__length__aux,axiom,
    ! [L2: list_nat,A3: nat] :
      ( ( foldr_nat_nat
        @ ^ [X: nat] : suc
        @ L2
        @ A3 )
      = ( plus_plus_nat @ A3 @ ( size_size_list_nat @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4639_foldr__length__aux,axiom,
    ! [L2: list_int,A3: nat] :
      ( ( foldr_int_nat
        @ ^ [X: int] : suc
        @ L2
        @ A3 )
      = ( plus_plus_nat @ A3 @ ( size_size_list_int @ L2 ) ) ) ).

% foldr_length_aux
thf(fact_4640_fst__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% fst_divmod
thf(fact_4641_fst__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( product_fst_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% fst_divmod
thf(fact_4642_word__of__int__2p,axiom,
    ! [N2: nat] :
      ( ( ring_17408606157368542149l_num1 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ).

% word_of_int_2p
thf(fact_4643_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4644_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4645_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_4646_rel__of__def,axiom,
    ( rel_of7001861737123029207T_VEBT
    = ( ^ [M3: nat > option1280308654545898343T_VEBT,P4: produc8398139464844984134T_VEBT > $o] :
          ( collec1047362574656026267T_VEBT
          @ ( produc2834603712688810931VEBT_o
            @ ^ [K3: nat,V3: produc4813437837504472865T_VEBT] :
                ( ( ( M3 @ K3 )
                  = ( some_P2407225848856114310T_VEBT @ V3 ) )
                & ( P4 @ ( produc1750349459881913976T_VEBT @ K3 @ V3 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_4647_rel__of__def,axiom,
    ( rel_of_num_num
    = ( ^ [M3: num > option_num,P4: product_prod_num_num > $o] :
          ( collec2230928802738392704um_num
          @ ( produc5703948589228662326_num_o
            @ ^ [K3: num,V3: num] :
                ( ( ( M3 @ K3 )
                  = ( some_num @ V3 ) )
                & ( P4 @ ( product_Pair_num_num @ K3 @ V3 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_4648_rel__of__def,axiom,
    ( rel_of_nat_num
    = ( ^ [M3: nat > option_num,P4: product_prod_nat_num > $o] :
          ( collec4100598399430923318at_num
          @ ( produc4927758841916487424_num_o
            @ ^ [K3: nat,V3: num] :
                ( ( ( M3 @ K3 )
                  = ( some_num @ V3 ) )
                & ( P4 @ ( product_Pair_nat_num @ K3 @ V3 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_4649_rel__of__def,axiom,
    ( rel_of_nat_nat
    = ( ^ [M3: nat > option_nat,P4: product_prod_nat_nat > $o] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [K3: nat,V3: nat] :
                ( ( ( M3 @ K3 )
                  = ( some_nat @ V3 ) )
                & ( P4 @ ( product_Pair_nat_nat @ K3 @ V3 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_4650_rel__of__def,axiom,
    ( rel_of_int_int
    = ( ^ [M3: int > option_int,P4: product_prod_int_int > $o] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [K3: int,V3: int] :
                ( ( ( M3 @ K3 )
                  = ( some_int @ V3 ) )
                & ( P4 @ ( product_Pair_int_int @ K3 @ V3 ) ) ) ) ) ) ) ).

% rel_of_def
thf(fact_4651_norm__divide__numeral,axiom,
    ! [A3: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A3 @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_4652_norm__divide__numeral,axiom,
    ! [A3: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A3 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_4653_norm__mult__numeral1,axiom,
    ! [W: num,A3: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A3 ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A3 ) ) ) ).

% norm_mult_numeral1
thf(fact_4654_norm__mult__numeral1,axiom,
    ! [W: num,A3: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A3 ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A3 ) ) ) ).

% norm_mult_numeral1
thf(fact_4655_norm__mult__numeral2,axiom,
    ! [A3: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A3 @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_4656_norm__mult__numeral2,axiom,
    ! [A3: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A3 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_4657_norm__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_4658_norm__le__zero__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
      = ( X2 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_4659_zero__less__norm__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
      = ( X2 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_4660_zero__less__norm__iff,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
      = ( X2 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_4661_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_4662_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_4663_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_4664_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_4665_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4666_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4667_norm__eq__zero,axiom,
    ! [X2: real] :
      ( ( ( real_V7735802525324610683m_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_4668_norm__eq__zero,axiom,
    ! [X2: complex] :
      ( ( ( real_V1022390504157884413omplex @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_4669_norm__not__less__zero,axiom,
    ! [X2: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_4670_norm__ge__zero,axiom,
    ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).

% norm_ge_zero
thf(fact_4671_norm__divide,axiom,
    ! [A3: real,B3: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ).

% norm_divide
thf(fact_4672_norm__divide,axiom,
    ! [A3: complex,B3: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ).

% norm_divide
thf(fact_4673_norm__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N2 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N2 ) ) ).

% norm_power
thf(fact_4674_norm__power,axiom,
    ! [X2: complex,N2: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N2 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N2 ) ) ).

% norm_power
thf(fact_4675_power__eq__imp__eq__norm,axiom,
    ! [W: real,N2: nat,Z: real] :
      ( ( ( power_power_real @ W @ N2 )
        = ( power_power_real @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4676_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N2: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N2 )
        = ( power_power_complex @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4677_nonzero__norm__divide,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A3 @ B3 ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A3 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4678_nonzero__norm__divide,axiom,
    ! [B3: complex,A3: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A3 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4679_norm__mult__less,axiom,
    ! [X2: real,R: real,Y2: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y2 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_4680_norm__mult__less,axiom,
    ! [X2: complex,R: real,Y2: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y2 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).

% norm_mult_less
thf(fact_4681_norm__triangle__lt,axiom,
    ! [X2: real,Y2: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y2 ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_4682_norm__triangle__lt,axiom,
    ! [X2: complex,Y2: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y2 ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_4683_norm__add__less,axiom,
    ! [X2: real,R: real,Y2: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y2 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_4684_norm__add__less,axiom,
    ! [X2: complex,R: real,Y2: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y2 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y2 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).

% norm_add_less
thf(fact_4685_norm__diff__triangle__less,axiom,
    ! [X2: real,Y2: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_4686_norm__diff__triangle__less,axiom,
    ! [X2: complex,Y2: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y2 @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_4687_norm__power__ineq,axiom,
    ! [X2: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_4688_norm__power__ineq,axiom,
    ! [X2: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_4689_power__eq__1__iff,axiom,
    ! [W: real,N2: nat] :
      ( ( ( power_power_real @ W @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_4690_power__eq__1__iff,axiom,
    ! [W: complex,N2: nat] :
      ( ( ( power_power_complex @ W @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_4691_square__norm__one,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X2 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4692_square__norm__one,axiom,
    ! [X2: complex] :
      ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X2 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_4693_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_4694_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_4695_lowi__def,axiom,
    ( vEBT_VEBT_lowi
    = ( ^ [X: nat,N: nat] : ( heap_Time_return_nat @ ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% lowi_def
thf(fact_4696_foldr__same,axiom,
    ! [Xs2: list_real,Y2: real] :
      ( ! [X3: real,Y3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y3 ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
           => ( X3 = Y2 ) )
       => ( ( foldr_real_real @ plus_plus_real @ Xs2 @ zero_zero_real )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Y2 ) ) ) ) ).

% foldr_same
thf(fact_4697_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat,Bound: nat,I: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4698_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > nat,Bound: nat,I: nat] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_Pr7570552894071451325sn_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_s6829681357464350627n_assn @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4699_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_real,F: real > nat,Bound: nat,I: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_real_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4700_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_o,F: $o > nat,Bound: nat,I: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_o_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4701_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_nat,F: nat > nat,Bound: nat,I: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4702_list__every__elemnt__bound__sum__bound,axiom,
    ! [Xs2: list_int,F: int > nat,Bound: nat,I: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_nat @ ( foldr_nat_nat @ plus_plus_nat @ ( map_int_nat @ F @ Xs2 ) @ I ) @ ( plus_plus_nat @ ( times_times_nat @ ( size_size_list_int @ Xs2 ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound
thf(fact_4703_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ B3 @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A3 @ one_one_int ) @ B3 @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( 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_4704_of__nat__code__if,axiom,
    ( semiri2565882477558803405uint32
    = ( ^ [N: nat] :
          ( if_uint32 @ ( N = zero_zero_nat ) @ zero_zero_uint32
          @ ( produc2417093276151063866uint32
            @ ^ [M3: nat,Q4: nat] : ( if_uint32 @ ( Q4 = zero_zero_nat ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ M3 ) ) @ ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ M3 ) ) @ one_one_uint32 ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4705_of__nat__code__if,axiom,
    ( semiri8819519690708144855l_num1
    = ( ^ [N: nat] :
          ( if_wor5778924947035936048l_num1 @ ( N = zero_zero_nat ) @ zero_z3563351764282998399l_num1
          @ ( produc6192303373133366212l_num1
            @ ^ [M3: nat,Q4: nat] : ( if_wor5778924947035936048l_num1 @ ( Q4 = zero_zero_nat ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ M3 ) ) @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ M3 ) ) @ one_on7727431528512463931l_num1 ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4706_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M3: nat,Q4: nat] : ( if_rat @ ( Q4 = 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4707_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M3: nat,Q4: nat] : ( if_real @ ( Q4 = 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4708_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M3: nat,Q4: nat] : ( if_int @ ( Q4 = 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4709_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M3: nat,Q4: nat] : ( if_nat @ ( Q4 = 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4710_of__nat__code__if,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N: nat] :
          ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger
          @ ( produc1830744345554046123nteger
            @ ^ [M3: nat,Q4: nat] : ( if_Code_integer @ ( Q4 = zero_zero_nat ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M3 ) ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ M3 ) ) @ one_one_Code_integer ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_4711_foldr0,axiom,
    ! [Xs2: list_real,C: real,D: real] :
      ( ( foldr_real_real @ plus_plus_real @ Xs2 @ ( plus_plus_real @ C @ D ) )
      = ( plus_plus_real @ ( foldr_real_real @ plus_plus_real @ Xs2 @ D ) @ C ) ) ).

% foldr0
thf(fact_4712_mod__mod__trivial,axiom,
    ! [A3: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_mod_trivial
thf(fact_4713_mod__mod__trivial,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_mod_trivial
thf(fact_4714_mod__mod__trivial,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_mod_trivial
thf(fact_4715_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% low_def
thf(fact_4716_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs: list_nat] : Xs ) ) ).

% map_ident
thf(fact_4717_bits__mod__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_0
thf(fact_4718_bits__mod__0,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_4719_bits__mod__0,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A3 )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_4720_bits__mod__0,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A3 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_4721_bits__mod__0,axiom,
    ! [A3: uint32] :
      ( ( modulo_modulo_uint32 @ zero_zero_uint32 @ A3 )
      = zero_zero_uint32 ) ).

% bits_mod_0
thf(fact_4722_mod__0,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A3 )
      = zero_zero_nat ) ).

% mod_0
thf(fact_4723_mod__0,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A3 )
      = zero_zero_int ) ).

% mod_0
thf(fact_4724_mod__0,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A3 )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_4725_mod__by__0,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% mod_by_0
thf(fact_4726_mod__by__0,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ A3 @ zero_zero_int )
      = A3 ) ).

% mod_by_0
thf(fact_4727_mod__by__0,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ zero_z3403309356797280102nteger )
      = A3 ) ).

% mod_by_0
thf(fact_4728_mod__self,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ A3 @ A3 )
      = zero_zero_nat ) ).

% mod_self
thf(fact_4729_mod__self,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ A3 @ A3 )
      = zero_zero_int ) ).

% mod_self
thf(fact_4730_mod__self,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ A3 )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_4731_mod__add__self1,axiom,
    ! [B3: nat,A3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B3 @ A3 ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_add_self1
thf(fact_4732_mod__add__self1,axiom,
    ! [B3: int,A3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B3 @ A3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_add_self1
thf(fact_4733_mod__add__self1,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B3 @ A3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_add_self1
thf(fact_4734_mod__add__self2,axiom,
    ! [A3: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_add_self2
thf(fact_4735_mod__add__self2,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_add_self2
thf(fact_4736_mod__add__self2,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_add_self2
thf(fact_4737_minus__mod__self2,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% minus_mod_self2
thf(fact_4738_minus__mod__self2,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% minus_mod_self2
thf(fact_4739_length__map,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_4740_length__map,axiom,
    ! [F: real > real,Xs2: list_real] :
      ( ( size_size_list_real @ ( map_real_real @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_4741_length__map,axiom,
    ! [F: $o > real,Xs2: list_o] :
      ( ( size_size_list_real @ ( map_o_real @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_4742_length__map,axiom,
    ! [F: nat > real,Xs2: list_nat] :
      ( ( size_size_list_real @ ( map_nat_real @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_4743_length__map,axiom,
    ! [F: int > real,Xs2: list_int] :
      ( ( size_size_list_real @ ( map_int_real @ F @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_map
thf(fact_4744_length__map,axiom,
    ! [F: real > $o,Xs2: list_real] :
      ( ( size_size_list_o @ ( map_real_o @ F @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_map
thf(fact_4745_length__map,axiom,
    ! [F: $o > $o,Xs2: list_o] :
      ( ( size_size_list_o @ ( map_o_o @ F @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_map
thf(fact_4746_length__map,axiom,
    ! [F: nat > $o,Xs2: list_nat] :
      ( ( size_size_list_o @ ( map_nat_o @ F @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_map
thf(fact_4747_length__map,axiom,
    ! [F: int > $o,Xs2: list_int] :
      ( ( size_size_list_o @ ( map_int_o @ F @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_map
thf(fact_4748_length__map,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT] :
      ( ( size_size_list_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_map
thf(fact_4749_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_4750_nat__mod__eq_H,axiom,
    ! [A3: nat,N2: nat] :
      ( ( ord_less_nat @ A3 @ N2 )
     => ( ( modulo_modulo_nat @ A3 @ N2 )
        = A3 ) ) ).

% nat_mod_eq'
thf(fact_4751_bits__mod__by__1,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ A3 @ one_on7727431528512463931l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_by_1
thf(fact_4752_bits__mod__by__1,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ A3 @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_4753_bits__mod__by__1,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ A3 @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_4754_bits__mod__by__1,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_4755_bits__mod__by__1,axiom,
    ! [A3: uint32] :
      ( ( modulo_modulo_uint32 @ A3 @ one_one_uint32 )
      = zero_zero_uint32 ) ).

% bits_mod_by_1
thf(fact_4756_mod__by__1,axiom,
    ! [A3: nat] :
      ( ( modulo_modulo_nat @ A3 @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_4757_mod__by__1,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ A3 @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_4758_mod__by__1,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_4759_mod__mult__self1__is__0,axiom,
    ! [B3: nat,A3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B3 @ A3 ) @ B3 )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_4760_mod__mult__self1__is__0,axiom,
    ! [B3: int,A3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B3 @ A3 ) @ B3 )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_4761_mod__mult__self1__is__0,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ B3 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_4762_mod__mult__self2__is__0,axiom,
    ! [A3: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_4763_mod__mult__self2__is__0,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_4764_mod__mult__self2__is__0,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ B3 )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_4765_bits__mod__div__trivial,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ B3 ) @ B3 )
      = zero_z3563351764282998399l_num1 ) ).

% bits_mod_div_trivial
thf(fact_4766_bits__mod__div__trivial,axiom,
    ! [A3: nat,B3: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_4767_bits__mod__div__trivial,axiom,
    ! [A3: int,B3: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_4768_bits__mod__div__trivial,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_4769_bits__mod__div__trivial,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A3 @ B3 ) @ B3 )
      = zero_zero_uint32 ) ).

% bits_mod_div_trivial
thf(fact_4770_mod__div__trivial,axiom,
    ! [A3: nat,B3: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_4771_mod__div__trivial,axiom,
    ! [A3: int,B3: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_4772_mod__div__trivial,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_4773_mod__mult__self4,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A3 ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_mult_self4
thf(fact_4774_mod__mult__self4,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_mult_self4
thf(fact_4775_mod__mult__self4,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ C ) @ A3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_mult_self4
thf(fact_4776_mod__mult__self3,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A3 ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_mult_self3
thf(fact_4777_mod__mult__self3,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A3 ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_mult_self3
thf(fact_4778_mod__mult__self3,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B3 ) @ A3 ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_mult_self3
thf(fact_4779_mod__mult__self2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_mult_self2
thf(fact_4780_mod__mult__self2,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ B3 @ C ) ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_mult_self2
thf(fact_4781_mod__mult__self2,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_mult_self2
thf(fact_4782_mod__mult__self1,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% mod_mult_self1
thf(fact_4783_mod__mult__self1,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( times_times_int @ C @ B3 ) ) @ B3 )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% mod_mult_self1
thf(fact_4784_mod__mult__self1,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) ) @ B3 )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% mod_mult_self1
thf(fact_4785_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_4786_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( divide_divide_nat @ M @ N2 ) ) ).

% fst_divmod_nat
thf(fact_4787_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ M @ N2 ) ) ).

% snd_divmod_nat
thf(fact_4788_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE8901447254227204932T_VEBT @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4789_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_VE7998069337340375161T_VEBT @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4790_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > nat] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBTi_nat @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4791_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > vEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE7029150624388687525_VEBTi @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4792_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > vEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_VEBT_VEBTi @ ( map_VE483055756984248624_VEBTi @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4793_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > int] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_int @ ( map_VEBT_VEBT_int @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4794_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,F: vEBT_VEBTi > int] :
      ( ( ord_less_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_int @ ( map_VEBT_VEBTi_int @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBTi @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4795_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4796_nth__map,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_VEBT_VEBT @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4797_nth__map,axiom,
    ! [N2: nat,Xs2: list_real,F: real > vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( map_real_VEBT_VEBT @ F @ Xs2 ) @ N2 )
        = ( F @ ( nth_real @ Xs2 @ N2 ) ) ) ) ).

% nth_map
thf(fact_4798_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_4799_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_4800_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_4801_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_4802_numeral__mod__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ K ) @ ( numera6620942414471956472nteger @ L2 ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ K @ L2 ) ) ) ).

% numeral_mod_numeral
thf(fact_4803_numeral__mod__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L2 ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ K @ L2 ) ) ) ).

% numeral_mod_numeral
thf(fact_4804_numeral__mod__numeral,axiom,
    ! [K: num,L2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ K @ L2 ) ) ) ).

% numeral_mod_numeral
thf(fact_4805_bits__one__mod__two__eq__one,axiom,
    ( ( modulo1504961113040953224l_num1 @ one_on7727431528512463931l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% bits_one_mod_two_eq_one
thf(fact_4806_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_4807_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_4808_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_4809_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_uint32 @ one_one_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_one_mod_two_eq_one
thf(fact_4810_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_4811_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_4812_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_4813_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_4814_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_4815_not__mod__2__eq__0__eq__1,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != zero_z3563351764282998399l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4816_not__mod__2__eq__0__eq__1,axiom,
    ! [A3: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != zero_zero_nat )
      = ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4817_not__mod__2__eq__0__eq__1,axiom,
    ! [A3: int] :
      ( ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != zero_zero_int )
      = ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4818_not__mod__2__eq__0__eq__1,axiom,
    ! [A3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4819_not__mod__2__eq__0__eq__1,axiom,
    ! [A3: uint32] :
      ( ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != zero_zero_uint32 )
      = ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4820_not__mod__2__eq__1__eq__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
       != one_on7727431528512463931l_num1 )
      = ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4821_not__mod__2__eq__1__eq__0,axiom,
    ! [A3: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != one_one_nat )
      = ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4822_not__mod__2__eq__1__eq__0,axiom,
    ! [A3: int] :
      ( ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4823_not__mod__2__eq__1__eq__0,axiom,
    ! [A3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4824_not__mod__2__eq__1__eq__0,axiom,
    ! [A3: uint32] :
      ( ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
       != one_one_uint32 )
      = ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4825_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
       != ( suc @ zero_zero_nat ) )
      = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% not_mod2_eq_Suc_0_eq_0
thf(fact_4826_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_4827_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_4828_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_4829_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_4830_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_4831_one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
      = ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ).

% one_mod_numeral
thf(fact_4832_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_4833_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_4834_list_Omap__ident,axiom,
    ! [T2: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T2 )
      = T2 ) ).

% list.map_ident
thf(fact_4835_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mod
thf(fact_4836_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mod
thf(fact_4837_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mod
thf(fact_4838_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > nat,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_VEBT_VEBT_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4839_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: vEBT_VEBT > real,Ys: list_VEBT_VEBT] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_VEBT_VEBT_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4840_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: real > nat,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_real_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4841_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: real > real,Ys: list_real] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_real_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4842_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: $o > nat,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_o_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4843_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: $o > real,Ys: list_o] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_o_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4844_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: nat > nat,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4845_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: nat > real,Ys: list_nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_nat_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4846_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,G: int > nat,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_nat @ F @ Xs2 )
        = ( map_int_nat @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4847_map__eq__imp__length__eq,axiom,
    ! [F: vEBT_VEBT > real,Xs2: list_VEBT_VEBT,G: int > real,Ys: list_int] :
      ( ( ( map_VEBT_VEBT_real @ F @ Xs2 )
        = ( map_int_real @ G @ Ys ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_4848_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > nat,L2: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_nat @ F @ L2 )
        = ( map_VEBT_VEBT_nat @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4849_map__eq__nth__eq,axiom,
    ! [F: vEBT_VEBT > real,L2: list_VEBT_VEBT,L4: list_VEBT_VEBT,I: nat] :
      ( ( ( map_VEBT_VEBT_real @ F @ L2 )
        = ( map_VEBT_VEBT_real @ F @ L4 ) )
     => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
        = ( F @ ( nth_VEBT_VEBT @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4850_map__eq__nth__eq,axiom,
    ! [F: nat > nat,L2: list_nat,L4: list_nat,I: nat] :
      ( ( ( map_nat_nat @ F @ L2 )
        = ( map_nat_nat @ F @ L4 ) )
     => ( ( F @ ( nth_nat @ L2 @ I ) )
        = ( F @ ( nth_nat @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4851_map__eq__nth__eq,axiom,
    ! [F: nat > $o,L2: list_nat,L4: list_nat,I: nat] :
      ( ( ( map_nat_o @ F @ L2 )
        = ( map_nat_o @ F @ L4 ) )
     => ( ( F @ ( nth_nat @ L2 @ I ) )
        = ( F @ ( nth_nat @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4852_map__eq__nth__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,L2: list_P8527749157015355191n_assn,L4: list_P8527749157015355191n_assn,I: nat] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ L2 )
        = ( map_Pr8991440229025900053n_assn @ F @ L4 ) )
     => ( ( F @ ( nth_Pr1769885009046257848n_assn @ L2 @ I ) )
        = ( F @ ( nth_Pr1769885009046257848n_assn @ L4 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_4853_mod__add__eq,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).

% mod_add_eq
thf(fact_4854_mod__add__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).

% mod_add_eq
thf(fact_4855_mod__add__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).

% mod_add_eq
thf(fact_4856_mod__add__cong,axiom,
    ! [A3: nat,C: nat,A5: nat,B3: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B3 @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4857_mod__add__cong,axiom,
    ! [A3: int,C: int,A5: int,B3: int,B4: int] :
      ( ( ( modulo_modulo_int @ A3 @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B3 @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4858_mod__add__cong,axiom,
    ! [A3: code_integer,C: code_integer,A5: code_integer,B3: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B3 @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4859_mod__add__left__eq,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B3 ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4860_mod__add__left__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4861_mod__add__left__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4862_mod__add__right__eq,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4863_mod__add__right__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A3 @ B3 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4864_mod__add__right__eq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4865_mod__mult__eq,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).

% mod_mult_eq
thf(fact_4866_mod__mult__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).

% mod_mult_eq
thf(fact_4867_mod__mult__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).

% mod_mult_eq
thf(fact_4868_mod__mult__cong,axiom,
    ! [A3: nat,C: nat,A5: nat,B3: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ C )
        = ( modulo_modulo_nat @ A5 @ C ) )
     => ( ( ( modulo_modulo_nat @ B3 @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4869_mod__mult__cong,axiom,
    ! [A3: int,C: int,A5: int,B3: int,B4: int] :
      ( ( ( modulo_modulo_int @ A3 @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B3 @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4870_mod__mult__cong,axiom,
    ! [A3: code_integer,C: code_integer,A5: code_integer,B3: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B3 @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_4871_mod__mult__mult2,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4872_mod__mult__mult2,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A3 @ B3 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4873_mod__mult__mult2,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ C ) ) ).

% mod_mult_mult2
thf(fact_4874_mult__mod__right,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ).

% mult_mod_right
thf(fact_4875_mult__mod__right,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) ) ) ).

% mult_mod_right
thf(fact_4876_mult__mod__right,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) ) ) ).

% mult_mod_right
thf(fact_4877_mod__mult__left__eq,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A3 @ C ) @ B3 ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4878_mod__mult__left__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4879_mod__mult__left__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_4880_mod__mult__right__eq,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4881_mod__mult__right__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4882_mod__mult__right__eq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_4883_mod__diff__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).

% mod_diff_eq
thf(fact_4884_mod__diff__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).

% mod_diff_eq
thf(fact_4885_mod__diff__cong,axiom,
    ! [A3: int,C: int,A5: int,B3: int,B4: int] :
      ( ( ( modulo_modulo_int @ A3 @ C )
        = ( modulo_modulo_int @ A5 @ C ) )
     => ( ( ( modulo_modulo_int @ B3 @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4886_mod__diff__cong,axiom,
    ! [A3: code_integer,C: code_integer,A5: code_integer,B3: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ C )
        = ( modulo364778990260209775nteger @ A5 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B3 @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A5 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4887_mod__diff__left__eq,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A3 @ C ) @ B3 ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4888_mod__diff__left__eq,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ B3 ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4889_mod__diff__right__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A3 @ B3 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4890_mod__diff__right__eq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4891_power__mod,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ N2 ) @ B3 )
      = ( modulo_modulo_nat @ ( power_power_nat @ A3 @ N2 ) @ B3 ) ) ).

% power_mod
thf(fact_4892_power__mod,axiom,
    ! [A3: int,B3: int,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A3 @ B3 ) @ N2 ) @ B3 )
      = ( modulo_modulo_int @ ( power_power_int @ A3 @ N2 ) @ B3 ) ) ).

% power_mod
thf(fact_4893_power__mod,axiom,
    ! [A3: code_integer,B3: code_integer,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ N2 ) @ B3 )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ B3 ) ) ).

% power_mod
thf(fact_4894_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_4895_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_4896_nat__mod__eq,axiom,
    ! [B3: nat,N2: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ N2 )
     => ( ( ( modulo_modulo_nat @ A3 @ N2 )
          = ( modulo_modulo_nat @ B3 @ N2 ) )
       => ( ( modulo_modulo_nat @ A3 @ N2 )
          = B3 ) ) ) ).

% nat_mod_eq
thf(fact_4897_mod__plus__right,axiom,
    ! [A3: nat,X2: nat,M: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ ( plus_plus_nat @ A3 @ X2 ) @ M )
        = ( modulo_modulo_nat @ ( plus_plus_nat @ B3 @ X2 ) @ M ) )
      = ( ( modulo_modulo_nat @ A3 @ M )
        = ( modulo_modulo_nat @ B3 @ M ) ) ) ).

% mod_plus_right
thf(fact_4898_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_4899_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N ) @ ( modulo_modulo_nat @ M3 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_4900_unique__quotient,axiom,
    ! [A3: int,B3: int,Q2: int,R: int,Q5: int,R5: int] :
      ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q5 @ R5 ) )
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
thf(fact_4901_unique__remainder,axiom,
    ! [A3: int,B3: int,Q2: int,R: int,Q5: int,R5: int] :
      ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q5 @ R5 ) )
       => ( R = R5 ) ) ) ).

% unique_remainder
thf(fact_4902_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ A3 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4903_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ A3 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4904_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A3 @ B3 ) @ A3 ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4905_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ B3 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4906_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4907_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ B3 ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4908_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4909_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4910_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4911_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4912_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4913_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4914_mod__eq__self__iff__div__eq__0,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ B3 )
        = A3 )
      = ( ( divide_divide_nat @ A3 @ B3 )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4915_mod__eq__self__iff__div__eq__0,axiom,
    ! [A3: int,B3: int] :
      ( ( ( modulo_modulo_int @ A3 @ B3 )
        = A3 )
      = ( ( divide_divide_int @ A3 @ B3 )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4916_mod__eq__self__iff__div__eq__0,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ B3 )
        = A3 )
      = ( ( divide6298287555418463151nteger @ A3 @ B3 )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4917_mod__eqE,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ( modulo_modulo_int @ A3 @ C )
        = ( modulo_modulo_int @ B3 @ C ) )
     => ~ ! [D2: int] :
            ( B3
           != ( plus_plus_int @ A3 @ ( times_times_int @ C @ D2 ) ) ) ) ).

% mod_eqE
thf(fact_4918_mod__eqE,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ C )
        = ( modulo364778990260209775nteger @ B3 @ C ) )
     => ~ ! [D2: code_integer] :
            ( B3
           != ( plus_p5714425477246183910nteger @ A3 @ ( times_3573771949741848930nteger @ C @ D2 ) ) ) ) ).

% mod_eqE
thf(fact_4919_div__add1__eq,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ C ) @ ( modulo_modulo_nat @ B3 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4920_div__add1__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ C ) @ ( modulo_modulo_int @ B3 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4921_div__add1__eq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ C ) @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4922_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_4923_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P2: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N4: nat] :
                ( ( ord_less_nat @ N4 @ P2 )
               => ( ( P @ N4 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_4924_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_4925_nat__mod__lem,axiom,
    ! [N2: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ B3 @ N2 )
        = ( ( modulo_modulo_nat @ B3 @ N2 )
          = B3 ) ) ) ).

% nat_mod_lem
thf(fact_4926_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_4927_word__rot__lem,axiom,
    ! [L2: nat,K: nat,D: nat,N2: nat] :
      ( ( ( plus_plus_nat @ L2 @ K )
        = ( plus_plus_nat @ D @ ( modulo_modulo_nat @ K @ L2 ) ) )
     => ( ( ord_less_nat @ N2 @ L2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ D @ N2 ) @ L2 )
          = N2 ) ) ) ).

% word_rot_lem
thf(fact_4928_nat__minus__mod,axiom,
    ! [N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ N2 @ ( modulo_modulo_nat @ N2 @ M ) ) @ M )
      = zero_zero_nat ) ).

% nat_minus_mod
thf(fact_4929_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_4930_mod__nat__sub,axiom,
    ! [X2: nat,Z: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( modulo_modulo_nat @ ( minus_minus_nat @ X2 @ Y2 ) @ Z )
        = ( minus_minus_nat @ X2 @ Y2 ) ) ) ).

% mod_nat_sub
thf(fact_4931_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_4932_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_4933_nat__minus__mod__plus__right,axiom,
    ! [N2: nat,X2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ X2 ) @ ( modulo_modulo_nat @ N2 @ M ) ) @ M )
      = ( modulo_modulo_nat @ X2 @ M ) ) ).

% nat_minus_mod_plus_right
thf(fact_4934_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_4935_nat__mod__eq__iff,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo_nat @ X2 @ N2 )
        = ( modulo_modulo_nat @ Y2 @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X2 @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y2 @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_4936_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
       => ( ( modulo364778990260209775nteger @ A3 @ B3 )
          = A3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4937_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ A3 @ B3 )
       => ( ( modulo_modulo_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4938_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ A3 @ B3 )
       => ( ( modulo_modulo_int @ A3 @ B3 )
          = A3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4939_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4940_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4941_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4942_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4943_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4944_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4945_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
      = zero_zero_nat ) ).

% cong_exp_iff_simps(1)
thf(fact_4946_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
      = zero_zero_int ) ).

% cong_exp_iff_simps(1)
thf(fact_4947_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_4948_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_4949_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_4950_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_4951_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4952_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4953_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4954_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4955_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4956_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_4957_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4958_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4959_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_4960_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4961_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4962_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q2: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_4963_cancel__div__mod__rules_I2_J,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ C )
      = ( plus_plus_nat @ A3 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4964_cancel__div__mod__rules_I2_J,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) @ C )
      = ( plus_plus_int @ A3 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4965_cancel__div__mod__rules_I2_J,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A3 @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4966_cancel__div__mod__rules_I1_J,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ C )
      = ( plus_plus_nat @ A3 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4967_cancel__div__mod__rules_I1_J,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) ) @ C )
      = ( plus_plus_int @ A3 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4968_cancel__div__mod__rules_I1_J,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A3 @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4969_mod__div__decomp,axiom,
    ! [A3: nat,B3: nat] :
      ( A3
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).

% mod_div_decomp
thf(fact_4970_mod__div__decomp,axiom,
    ! [A3: int,B3: int] :
      ( A3
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% mod_div_decomp
thf(fact_4971_mod__div__decomp,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( A3
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).

% mod_div_decomp
thf(fact_4972_mod__div__decomp,axiom,
    ! [A3: uint32,B3: uint32] :
      ( A3
      = ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_uint32 @ A3 @ B3 ) ) ) ).

% mod_div_decomp
thf(fact_4973_div__mult__mod__eq,axiom,
    ! [A3: nat,B3: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = A3 ) ).

% div_mult_mod_eq
thf(fact_4974_div__mult__mod__eq,axiom,
    ! [A3: int,B3: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_int @ A3 @ B3 ) )
      = A3 ) ).

% div_mult_mod_eq
thf(fact_4975_div__mult__mod__eq,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
      = A3 ) ).

% div_mult_mod_eq
thf(fact_4976_div__mult__mod__eq,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A3 @ B3 ) @ B3 ) @ ( modulo_modulo_uint32 @ A3 @ B3 ) )
      = A3 ) ).

% div_mult_mod_eq
thf(fact_4977_mod__div__mult__eq,axiom,
    ! [A3: nat,B3: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) )
      = A3 ) ).

% mod_div_mult_eq
thf(fact_4978_mod__div__mult__eq,axiom,
    ! [A3: int,B3: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A3 @ B3 ) @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) )
      = A3 ) ).

% mod_div_mult_eq
thf(fact_4979_mod__div__mult__eq,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) )
      = A3 ) ).

% mod_div_mult_eq
thf(fact_4980_mod__div__mult__eq,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ B3 ) @ ( times_times_uint32 @ ( divide_divide_uint32 @ A3 @ B3 ) @ B3 ) )
      = A3 ) ).

% mod_div_mult_eq
thf(fact_4981_mod__mult__div__eq,axiom,
    ! [A3: nat,B3: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) )
      = A3 ) ).

% mod_mult_div_eq
thf(fact_4982_mod__mult__div__eq,axiom,
    ! [A3: int,B3: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A3 @ B3 ) @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) )
      = A3 ) ).

% mod_mult_div_eq
thf(fact_4983_mod__mult__div__eq,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) )
      = A3 ) ).

% mod_mult_div_eq
thf(fact_4984_mod__mult__div__eq,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ B3 ) @ ( times_times_uint32 @ B3 @ ( divide_divide_uint32 @ A3 @ B3 ) ) )
      = A3 ) ).

% mod_mult_div_eq
thf(fact_4985_mult__div__mod__eq,axiom,
    ! [B3: nat,A3: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = A3 ) ).

% mult_div_mod_eq
thf(fact_4986_mult__div__mod__eq,axiom,
    ! [B3: int,A3: int] :
      ( ( plus_plus_int @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) @ ( modulo_modulo_int @ A3 @ B3 ) )
      = A3 ) ).

% mult_div_mod_eq
thf(fact_4987_mult__div__mod__eq,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
      = A3 ) ).

% mult_div_mod_eq
thf(fact_4988_mult__div__mod__eq,axiom,
    ! [B3: uint32,A3: uint32] :
      ( ( plus_plus_uint32 @ ( times_times_uint32 @ B3 @ ( divide_divide_uint32 @ A3 @ B3 ) ) @ ( modulo_modulo_uint32 @ A3 @ B3 ) )
      = A3 ) ).

% mult_div_mod_eq
thf(fact_4989_div__mult1__eq,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A3 @ ( modulo_modulo_nat @ B3 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4990_div__mult1__eq,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A3 @ ( modulo_modulo_int @ B3 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4991_div__mult1__eq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ ( modulo364778990260209775nteger @ B3 @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4992_minus__div__mult__eq__mod,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% minus_div_mult_eq_mod
thf(fact_4993_minus__div__mult__eq__mod,axiom,
    ! [A3: int,B3: int] :
      ( ( minus_minus_int @ A3 @ ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% minus_div_mult_eq_mod
thf(fact_4994_minus__div__mult__eq__mod,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ A3 @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% minus_div_mult_eq_mod
thf(fact_4995_minus__div__mult__eq__mod,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( minus_minus_uint32 @ A3 @ ( times_times_uint32 @ ( divide_divide_uint32 @ A3 @ B3 ) @ B3 ) )
      = ( modulo_modulo_uint32 @ A3 @ B3 ) ) ).

% minus_div_mult_eq_mod
thf(fact_4996_minus__mod__eq__div__mult,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ B3 ) ) ).

% minus_mod_eq_div_mult
thf(fact_4997_minus__mod__eq__div__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) )
      = ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 ) ) ).

% minus_mod_eq_div_mult
thf(fact_4998_minus__mod__eq__div__mult,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ B3 ) ) ).

% minus_mod_eq_div_mult
thf(fact_4999_minus__mod__eq__div__mult,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( minus_minus_uint32 @ A3 @ ( modulo_modulo_uint32 @ A3 @ B3 ) )
      = ( times_times_uint32 @ ( divide_divide_uint32 @ A3 @ B3 ) @ B3 ) ) ).

% minus_mod_eq_div_mult
thf(fact_5000_minus__mod__eq__mult__div,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5001_minus__mod__eq__mult__div,axiom,
    ! [A3: int,B3: int] :
      ( ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) )
      = ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5002_minus__mod__eq__mult__div,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
      = ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5003_minus__mod__eq__mult__div,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( minus_minus_uint32 @ A3 @ ( modulo_modulo_uint32 @ A3 @ B3 ) )
      = ( times_times_uint32 @ B3 @ ( divide_divide_uint32 @ A3 @ B3 ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_5004_minus__mult__div__eq__mod,axiom,
    ! [A3: nat,B3: nat] :
      ( ( minus_minus_nat @ A3 @ ( times_times_nat @ B3 @ ( divide_divide_nat @ A3 @ B3 ) ) )
      = ( modulo_modulo_nat @ A3 @ B3 ) ) ).

% minus_mult_div_eq_mod
thf(fact_5005_minus__mult__div__eq__mod,axiom,
    ! [A3: int,B3: int] :
      ( ( minus_minus_int @ A3 @ ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) ) )
      = ( modulo_modulo_int @ A3 @ B3 ) ) ).

% minus_mult_div_eq_mod
thf(fact_5006_minus__mult__div__eq__mod,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( minus_8373710615458151222nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) )
      = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ).

% minus_mult_div_eq_mod
thf(fact_5007_minus__mult__div__eq__mod,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( minus_minus_uint32 @ A3 @ ( times_times_uint32 @ B3 @ ( divide_divide_uint32 @ A3 @ B3 ) ) )
      = ( modulo_modulo_uint32 @ A3 @ B3 ) ) ).

% minus_mult_div_eq_mod
thf(fact_5008_zmde,axiom,
    ! [B3: int,A3: int] :
      ( ( times_times_int @ B3 @ ( divide_divide_int @ A3 @ B3 ) )
      = ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% zmde
thf(fact_5009_zmde,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ A3 @ B3 ) )
      = ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).

% zmde
thf(fact_5010_zmde,axiom,
    ! [B3: uint32,A3: uint32] :
      ( ( times_times_uint32 @ B3 @ ( divide_divide_uint32 @ A3 @ B3 ) )
      = ( minus_minus_uint32 @ A3 @ ( modulo_modulo_uint32 @ A3 @ B3 ) ) ) ).

% zmde
thf(fact_5011_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_5012_div__less__mono,axiom,
    ! [A4: nat,B6: nat,N2: nat] :
      ( ( ord_less_nat @ A4 @ B6 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A4 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B6 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N2 ) @ ( divide_divide_nat @ B6 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_5013_mod__nat__add,axiom,
    ! [X2: nat,Z: nat,Y2: nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( ord_less_nat @ Y2 @ Z )
       => ( ( ( ord_less_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ Z )
              = ( plus_plus_nat @ X2 @ Y2 ) ) )
          & ( ~ ( ord_less_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ Z )
           => ( ( modulo_modulo_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ Z )
              = ( minus_minus_nat @ ( plus_plus_nat @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ).

% mod_nat_add
thf(fact_5014_nat__mod__eq__lemma,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( ( modulo_modulo_nat @ X2 @ N2 )
        = ( modulo_modulo_nat @ Y2 @ N2 ) )
     => ( ( ord_less_eq_nat @ Y2 @ X2 )
       => ? [Q3: nat] :
            ( X2
            = ( plus_plus_nat @ Y2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_5015_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S3: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_5016_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q2 )
        = ( modulo_modulo_nat @ N2 @ Q2 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_5017_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_5018_div__mod__decomp,axiom,
    ! [A4: nat,N2: nat] :
      ( A4
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A4 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A4 @ N2 ) ) ) ).

% div_mod_decomp
thf(fact_5019_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M3: nat,N: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N ) @ N ) ) ) ) ).

% modulo_nat_def
thf(fact_5020_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_5021_snd__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( produc6174133586879617921nteger @ ( unique3479559517661332726nteger @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% snd_divmod
thf(fact_5022_snd__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% snd_divmod
thf(fact_5023_snd__divmod,axiom,
    ! [M: num,N2: num] :
      ( ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% snd_divmod
thf(fact_5024_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_5025_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_5026_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_5027_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_5028_map__upd__eq,axiom,
    ! [I: nat,L2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn,X2: produc6575502325842934193n_assn] :
      ( ( ( ord_less_nat @ I @ ( size_s6829681357464350627n_assn @ L2 ) )
       => ( ( F @ ( nth_Pr1769885009046257848n_assn @ L2 @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_Pr8991440229025900053n_assn @ F @ ( list_u4534839942911652127n_assn @ L2 @ I @ X2 ) )
        = ( map_Pr8991440229025900053n_assn @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5029_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F: vEBT_VEBT > nat,X2: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_VEBT_VEBT_nat @ F @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X2 ) )
        = ( map_VEBT_VEBT_nat @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5030_map__upd__eq,axiom,
    ! [I: nat,L2: list_VEBT_VEBT,F: vEBT_VEBT > real,X2: vEBT_VEBT] :
      ( ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ L2 ) )
       => ( ( F @ ( nth_VEBT_VEBT @ L2 @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_VEBT_VEBT_real @ F @ ( list_u1324408373059187874T_VEBT @ L2 @ I @ X2 ) )
        = ( map_VEBT_VEBT_real @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5031_map__upd__eq,axiom,
    ! [I: nat,L2: list_nat,F: nat > nat,X2: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
       => ( ( F @ ( nth_nat @ L2 @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_nat_nat @ F @ ( list_update_nat @ L2 @ I @ X2 ) )
        = ( map_nat_nat @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5032_map__upd__eq,axiom,
    ! [I: nat,L2: list_nat,F: nat > $o,X2: nat] :
      ( ( ( ord_less_nat @ I @ ( size_size_list_nat @ L2 ) )
       => ( ( F @ ( nth_nat @ L2 @ I ) )
          = ( F @ X2 ) ) )
     => ( ( map_nat_o @ F @ ( list_update_nat @ L2 @ I @ X2 ) )
        = ( map_nat_o @ F @ L2 ) ) ) ).

% map_upd_eq
thf(fact_5033_mod__mult2__eq_H,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A3 @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A3 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5034_mod__mult2__eq_H,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A3 @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5035_mod__mult2__eq_H,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_5036_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5037_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5038_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M3: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5039_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5040_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5041_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_5042_mod__lemma,axiom,
    ! [C: nat,R: nat,B3: nat,Q2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( ( ord_less_nat @ R @ B3 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ Q2 @ C ) ) @ R ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).

% mod_lemma
thf(fact_5043_real__of__nat__div__aux,axiom,
    ! [X2: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X2 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X2 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_5044_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_5045_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B3 @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5046_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5047_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_5048_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_5049_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_5050_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_5051_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5052_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5053_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5054_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_5055_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M3: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5056_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_5057_divmod__digit__0_I2_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5058_divmod__digit__0_I2_J,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5059_divmod__digit__0_I2_J,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
          = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_5060_bits__stable__imp__add__self,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = A3 )
     => ( ( plus_p361126936061061375l_num1 @ A3 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% bits_stable_imp_add_self
thf(fact_5061_bits__stable__imp__add__self,axiom,
    ! [A3: nat] :
      ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A3 )
     => ( ( plus_plus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_nat ) ) ).

% bits_stable_imp_add_self
thf(fact_5062_bits__stable__imp__add__self,axiom,
    ! [A3: int] :
      ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A3 )
     => ( ( plus_plus_int @ A3 @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = zero_zero_int ) ) ).

% bits_stable_imp_add_self
thf(fact_5063_bits__stable__imp__add__self,axiom,
    ! [A3: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A3 )
     => ( ( plus_p5714425477246183910nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_5064_bits__stable__imp__add__self,axiom,
    ! [A3: uint32] :
      ( ( ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = A3 )
     => ( ( plus_plus_uint32 @ A3 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
        = zero_zero_uint32 ) ) ).

% bits_stable_imp_add_self
thf(fact_5065_div__exp__mod__exp__eq,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: nat,M: nat] :
      ( ( modulo1504961113040953224l_num1 @ ( divide1791077408188789448l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) )
      = ( divide1791077408188789448l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5066_div__exp__mod__exp__eq,axiom,
    ! [A3: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5067_div__exp__mod__exp__eq,axiom,
    ! [A3: int,N2: nat,M: nat] :
      ( ( modulo_modulo_int @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5068_div__exp__mod__exp__eq,axiom,
    ! [A3: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5069_div__exp__mod__exp__eq,axiom,
    ! [A3: uint32,N2: nat,M: nat] :
      ( ( modulo_modulo_uint32 @ ( divide_divide_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) )
      = ( divide_divide_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_5070_power__mod__div,axiom,
    ! [X2: nat,N2: nat,M: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
      = ( modulo_modulo_nat @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% power_mod_div
thf(fact_5071_verit__le__mono__div,axiom,
    ! [A4: nat,B6: nat,N2: nat] :
      ( ( ord_less_nat @ A4 @ B6 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B6 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B6 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_5072_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt2 @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary4 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList2 ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.cnt'.simps(2)
thf(fact_5073_divmod__digit__0_I1_J,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B3 )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5074_divmod__digit__0_I1_J,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5075_divmod__digit__0_I1_J,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
          = ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_5076_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo1504961113040953224l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_7065122842183080059l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5077_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_nat @ ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5078_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A3: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_int @ ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5079_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A3: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5080_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A3: uint32] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_uint32 @ ( times_times_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_times_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_5081_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2 != one_one_nat ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.cnt'.elims
thf(fact_5082_mod__double__modulus,axiom,
    ! [M: code_integer,X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
       => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X2 @ M ) )
          | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5083_mod__double__modulus,axiom,
    ! [M: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
       => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X2 @ M ) )
          | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5084_mod__double__modulus,axiom,
    ! [M: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X2 @ M ) )
          | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_5085_divmod__digit__1_I2_J,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5086_divmod__digit__1_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo_modulo_nat @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5087_divmod__digit__1_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
            = ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_5088_set__bit__Suc,axiom,
    ! [N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( bit_se4894374433684937756l_num1 @ ( suc @ N2 ) @ A3 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5089_set__bit__Suc,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A3 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5090_set__bit__Suc,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( bit_se6647067497041451410uint32 @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ N2 @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5091_set__bit__Suc,axiom,
    ! [N2: nat,A3: int] :
      ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5092_set__bit__Suc,axiom,
    ! [N2: nat,A3: nat] :
      ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_5093_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_5094_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_VEBT_space2 @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary4 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList2 ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space'.simps(2)
thf(fact_5095_divmod__digit__1_I1_J,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
       => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5096_divmod__digit__1_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ B3 )
       => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5097_divmod__digit__1_I1_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_int )
            = ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_5098_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space'.elims
thf(fact_5099_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_VEBT_space @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary4 ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList2 ) @ zero_zero_nat ) ) ) ).

% VEBT_internal.space.simps(2)
thf(fact_5100_VEBT__internal_Ospace_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
               != ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList3 ) @ zero_zero_nat ) ) ) ) ) ) ).

% VEBT_internal.space.elims
thf(fact_5101_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B3 )
     => ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q2 @ R ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( 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_5102_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_5103_div__half__nat,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ( ( product_Pair_nat_nat @ ( divide_divide_nat @ X2 @ Y2 ) @ ( modulo_modulo_nat @ X2 @ Y2 ) )
        = ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ Y2 @ ( minus_minus_nat @ X2 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) @ Y2 ) ) ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( minus_minus_nat @ X2 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) @ Y2 ) ) @ Y2 ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) @ ( minus_minus_nat @ X2 @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) @ Y2 ) ) ) ) ) ) ).

% div_half_nat
thf(fact_5104_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_5105_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Bound: real,I: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5106_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > real,Bound: real,I: real] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_Pr5018725648611240729n_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_s6829681357464350627n_assn @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5107_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_real,F: real > real,Bound: real,I: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_real @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5108_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_o,F: $o > real,Bound: real,I: real] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_o_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_o @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5109_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_nat,F: nat > real,Bound: real,I: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_nat @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5110_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [Xs2: list_int,F: int > real,Bound: real,I: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ Bound ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ I ) @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( size_size_list_int @ Xs2 ) ) @ Bound ) @ I ) ) ) ).

% list_every_elemnt_bound_sum_bound_real
thf(fact_5111_real__nat__list,axiom,
    ! [F: vEBT_VEBT > nat,Xs2: list_VEBT_VEBT,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_VEBT_VEBT_real
          @ ^ [X: vEBT_VEBT] : ( semiri5074537144036343181t_real @ ( F @ X ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_5112_real__nat__list,axiom,
    ! [F: nat > nat,Xs2: list_nat,C: nat] :
      ( ( semiri5074537144036343181t_real @ ( foldr_nat_nat @ plus_plus_nat @ ( map_nat_nat @ F @ Xs2 ) @ C ) )
      = ( foldr_real_real @ plus_plus_real
        @ ( map_nat_real
          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
          @ Xs2 )
        @ ( semiri5074537144036343181t_real @ C ) ) ) ).

% real_nat_list
thf(fact_5113_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,C: real,G: vEBT_VEBT > real,D: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5114_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,C: real,G: nat > real,D: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5115_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_real,F: real > real,C: real,G: real > real,D: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5116_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_int,F: int > real,C: real,G: int > real,D: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5117_f__g__map__foldr__bound,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > real,C: real,G: produc6575502325842934193n_assn > real,D: real] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ord_less_eq_real @ ( F @ X3 ) @ ( times_times_real @ C @ ( G @ X3 ) ) ) )
     => ( ord_less_eq_real @ ( foldr_real_real @ plus_plus_real @ ( map_Pr5018725648611240729n_real @ F @ Xs2 ) @ D ) @ ( plus_plus_real @ ( times_times_real @ C @ ( foldr_real_real @ plus_plus_real @ ( map_Pr5018725648611240729n_real @ G @ Xs2 ) @ zero_zero_real ) ) @ D ) ) ) ).

% f_g_map_foldr_bound
thf(fact_5118_round__unique,axiom,
    ! [X2: real,Y2: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y2 ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y2 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X2 )
          = Y2 ) ) ) ).

% round_unique
thf(fact_5119_round__unique,axiom,
    ! [X2: rat,Y2: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y2 ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y2 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X2 )
          = Y2 ) ) ) ).

% round_unique
thf(fact_5120_mult__le__cancel__iff1,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ Z ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5121_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
        = ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5122_mult__le__cancel__iff1,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
        = ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_5123_mod__word__self,axiom,
    ! [W: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ W @ W )
      = zero_z3563351764282998399l_num1 ) ).

% mod_word_self
thf(fact_5124_listsum__bound,axiom,
    ! [Xs2: list_VEBT_VEBT,F: vEBT_VEBT > real,Y2: real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y2 @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_5125_listsum__bound,axiom,
    ! [Xs2: list_nat,F: nat > real,Y2: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y2 @ ( foldr_real_real @ plus_plus_real @ ( map_nat_real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_5126_listsum__bound,axiom,
    ! [Xs2: list_real,F: real > real,Y2: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y2 @ ( foldr_real_real @ plus_plus_real @ ( map_real_real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_5127_listsum__bound,axiom,
    ! [Xs2: list_int,F: int > real,Y2: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y2 @ ( foldr_real_real @ plus_plus_real @ ( map_int_real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_5128_listsum__bound,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > real,Y2: real] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ Y2 @ ( foldr_real_real @ plus_plus_real @ ( map_Pr5018725648611240729n_real @ F @ Xs2 ) @ Y2 ) ) ) ).

% listsum_bound
thf(fact_5129_map__fst__mk__snd,axiom,
    ! [K: num,L2: list_num] :
      ( ( map_Pr454908937103039467um_num @ product_fst_num_num
        @ ( map_nu2851882102140640437um_num
          @ ^ [X: num] : ( product_Pair_num_num @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5130_map__fst__mk__snd,axiom,
    ! [K: produc4813437837504472865T_VEBT,L2: list_nat] :
      ( ( map_Pr3018521781701129308BT_nat @ produc758997459209783180T_VEBT
        @ ( map_na3322839687800594908T_VEBT
          @ ^ [X: nat] : ( produc1750349459881913976T_VEBT @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5131_map__fst__mk__snd,axiom,
    ! [K: num,L2: list_nat] :
      ( ( map_Pr5956769322976601943um_nat @ product_fst_nat_num
        @ ( map_na8006665559001981237at_num
          @ ^ [X: nat] : ( product_Pair_nat_num @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5132_map__fst__mk__snd,axiom,
    ! [K: nat,L2: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5133_map__fst__mk__snd,axiom,
    ! [K: int,L2: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int
        @ ( map_in7157766398909135175nt_int
          @ ^ [X: int] : ( product_Pair_int_int @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5134_map__fst__mk__snd,axiom,
    ! [K: assn,L2: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn
        @ ( map_as2373307505041272643n_assn
          @ ^ [X: assn] : ( produc118845697133431529n_assn @ X @ K )
          @ L2 ) )
      = L2 ) ).

% map_fst_mk_snd
thf(fact_5135_map__snd__mk__fst,axiom,
    ! [K: num,L2: list_num] :
      ( ( map_Pr454908937103039467um_num @ product_snd_num_num @ ( map_nu2851882102140640437um_num @ ( product_Pair_num_num @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5136_map__snd__mk__fst,axiom,
    ! [K: nat,L2: list_P6730324909620535719T_VEBT] :
      ( ( map_Pr3651661567648760725T_VEBT @ produc2084898568784432842T_VEBT @ ( map_Pr7909041933469137955T_VEBT @ ( produc1750349459881913976T_VEBT @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5137_map__snd__mk__fst,axiom,
    ! [K: nat,L2: list_num] :
      ( ( map_Pr2514101109132380577um_num @ product_snd_nat_num @ ( map_nu4721551698833171051at_num @ ( product_Pair_nat_num @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5138_map__snd__mk__fst,axiom,
    ! [K: nat,L2: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5139_map__snd__mk__fst,axiom,
    ! [K: int,L2: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5140_map__snd__mk__fst,axiom,
    ! [K: assn,L2: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ ( map_as2373307505041272643n_assn @ ( produc118845697133431529n_assn @ K ) @ L2 ) )
      = L2 ) ).

% map_snd_mk_fst
thf(fact_5141_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_5142_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_5143_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_5144_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_5145_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5146_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_5147_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_5148_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_5149_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_5150_one__mod__exp__eq__one,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = one_one_int ) ).

% one_mod_exp_eq_one
thf(fact_5151_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_5152_word__mod__by__0,axiom,
    ! [K: word_N3645301735248828278l_num1] :
      ( ( modulo1504961113040953224l_num1 @ K @ zero_z3563351764282998399l_num1 )
      = K ) ).

% word_mod_by_0
thf(fact_5153_zmod__helper,axiom,
    ! [N2: int,M: int,K: int,A3: int] :
      ( ( ( modulo_modulo_int @ N2 @ M )
        = K )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ N2 @ A3 ) @ M )
        = ( modulo_modulo_int @ ( plus_plus_int @ K @ A3 ) @ M ) ) ) ).

% zmod_helper
thf(fact_5154_pair__list__eqI,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Xs2 )
        = ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Ys ) )
     => ( ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Xs2 )
          = ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5155_pair__list__eqI,axiom,
    ! [Xs2: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
      ( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs2 )
        = ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
     => ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs2 )
          = ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5156_pair__list__eqI,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ Xs2 )
        = ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ Ys ) )
     => ( ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ Xs2 )
          = ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_5157_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_5158_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_5159_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_5160_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_5161_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q4: int] :
            ( M
            = ( times_times_int @ D @ Q4 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_5162_word__mod__less__divisor,axiom,
    ! [N2: word_N3645301735248828278l_num1,M: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ zero_z3563351764282998399l_num1 @ N2 )
     => ( ord_le750835935415966154l_num1 @ ( modulo1504961113040953224l_num1 @ M @ N2 ) @ N2 ) ) ).

% word_mod_less_divisor
thf(fact_5163_zmod__int,axiom,
    ! [A3: nat,B3: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A3 @ B3 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).

% zmod_int
thf(fact_5164_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_5165_round__mono,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ X2 @ Y2 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y2 ) ) ) ).

% round_mono
thf(fact_5166_int__mod__ge,axiom,
    ! [A3: int,N2: int] :
      ( ( ord_less_int @ A3 @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ A3 @ ( modulo_modulo_int @ A3 @ N2 ) ) ) ) ).

% int_mod_ge
thf(fact_5167_int__mod__lem,axiom,
    ! [N2: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ N2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ B3 )
          & ( ord_less_int @ B3 @ N2 ) )
        = ( ( modulo_modulo_int @ B3 @ N2 )
          = B3 ) ) ) ).

% int_mod_lem
thf(fact_5168_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_5169_int__mod__eq,axiom,
    ! [B3: int,N2: int,A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B3 )
     => ( ( ord_less_int @ B3 @ N2 )
       => ( ( ( modulo_modulo_int @ A3 @ N2 )
            = ( modulo_modulo_int @ B3 @ N2 ) )
         => ( ( modulo_modulo_int @ A3 @ N2 )
            = B3 ) ) ) ) ).

% int_mod_eq
thf(fact_5170_pos__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ B3 ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 ) ) ) ).

% pos_mod_conj
thf(fact_5171_neg__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A3 @ B3 ) @ zero_zero_int )
        & ( ord_less_int @ B3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).

% neg_mod_conj
thf(fact_5172_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_5173_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_5174_int__mod__le_H,axiom,
    ! [B3: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ B3 @ N2 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ B3 @ N2 ) @ ( minus_minus_int @ B3 @ N2 ) ) ) ).

% int_mod_le'
thf(fact_5175_nonneg__mod__div,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ B3 ) )
          & ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ) ).

% nonneg_mod_div
thf(fact_5176_zdiv__mono__strict,axiom,
    ! [A4: int,B6: int,N2: int] :
      ( ( ord_less_int @ A4 @ B6 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A4 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B6 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A4 @ N2 ) @ ( divide_divide_int @ B6 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_5177_div__mod__decomp__int,axiom,
    ! [A4: int,N2: int] :
      ( A4
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A4 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_5178_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M3: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_5179_mod__div__equality__div__eq,axiom,
    ! [A3: int,B3: int] :
      ( ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ B3 )
      = ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% mod_div_equality_div_eq
thf(fact_5180_ceiling__ge__round,axiom,
    ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).

% ceiling_ge_round
thf(fact_5181_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_5182_pos__mod__bound2,axiom,
    ! [A3: int] : ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% pos_mod_bound2
thf(fact_5183_int__mod__ge_H,axiom,
    ! [B3: int,N2: int] :
      ( ( ord_less_int @ B3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ B3 @ N2 ) @ ( modulo_modulo_int @ B3 @ N2 ) ) ) ) ).

% int_mod_ge'
thf(fact_5184_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_5185_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_5186_real__of__int__div__aux,axiom,
    ! [X2: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X2 ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X2 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X2 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_5187_pos__mod__sign2,axiom,
    ! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% pos_mod_sign2
thf(fact_5188_nmod2,axiom,
    ! [N2: int] :
      ( ( ( modulo_modulo_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int )
      | ( ( modulo_modulo_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% nmod2
thf(fact_5189_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
       != one_one_int )
      = ( ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% mod_2_neq_1_eq_eq_0
thf(fact_5190_mod__exp__less__eq__exp,axiom,
    ! [A3: int,N2: nat] : ( ord_less_int @ ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% mod_exp_less_eq_exp
thf(fact_5191_mod__power__lem,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A3 )
     => ( ( ( ord_less_eq_nat @ M @ N2 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ A3 @ M ) )
            = zero_zero_int ) )
        & ( ~ ( ord_less_eq_nat @ M @ N2 )
         => ( ( modulo_modulo_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ A3 @ M ) )
            = ( power_power_int @ A3 @ N2 ) ) ) ) ) ).

% mod_power_lem
thf(fact_5192_int__mod__pos__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R: int] :
      ( ( A3
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R )
       => ( ( ord_less_int @ R @ B3 )
         => ( ( modulo_modulo_int @ A3 @ B3 )
            = R ) ) ) ) ).

% int_mod_pos_eq
thf(fact_5193_int__mod__neg__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R: int] :
      ( ( A3
        = ( plus_plus_int @ ( times_times_int @ B3 @ Q2 ) @ R ) )
     => ( ( ord_less_eq_int @ R @ zero_zero_int )
       => ( ( ord_less_int @ B3 @ R )
         => ( ( modulo_modulo_int @ A3 @ B3 )
            = R ) ) ) ) ).

% int_mod_neg_eq
thf(fact_5194_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_5195_mod__sub__if__z,axiom,
    ! [X2: int,Z: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_eq_int @ Y2 @ X2 )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X2 @ Y2 ) @ Z )
                    = ( minus_minus_int @ X2 @ Y2 ) ) )
                & ( ~ ( ord_less_eq_int @ Y2 @ X2 )
                 => ( ( modulo_modulo_int @ ( minus_minus_int @ X2 @ Y2 ) @ Z )
                    = ( plus_plus_int @ ( minus_minus_int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_sub_if_z
thf(fact_5196_mod__add__if__z,axiom,
    ! [X2: int,Z: int,Y2: int] :
      ( ( ord_less_int @ X2 @ Z )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
           => ( ( ord_less_eq_int @ zero_zero_int @ Z )
             => ( ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X2 @ Y2 ) @ Z )
                    = ( plus_plus_int @ X2 @ Y2 ) ) )
                & ( ~ ( ord_less_int @ ( plus_plus_int @ X2 @ Y2 ) @ Z )
                 => ( ( modulo_modulo_int @ ( plus_plus_int @ X2 @ Y2 ) @ Z )
                    = ( minus_minus_int @ ( plus_plus_int @ X2 @ Y2 ) @ Z ) ) ) ) ) ) ) ) ) ).

% mod_add_if_z
thf(fact_5197_zmod__zmult2__eq,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_5198_axxmod2,axiom,
    ! [X2: int] :
      ( ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ X2 ) @ X2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int )
      & ( ( modulo_modulo_int @ ( plus_plus_int @ ( plus_plus_int @ zero_zero_int @ X2 ) @ X2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% axxmod2
thf(fact_5199_z1pmod2,axiom,
    ! [B3: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = one_one_int ) ).

% z1pmod2
thf(fact_5200_verit__le__mono__div__int,axiom,
    ! [A4: int,B6: int,N2: int] :
      ( ( ord_less_int @ A4 @ B6 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A4 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B6 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B6 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_5201_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_5202_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I4: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
             => ( P @ I4 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_5203_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info4: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_VEBT_cnt @ ( vEBT_Node @ Info4 @ Deg4 @ TreeList2 @ Summary4 ) )
      = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary4 ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList2 ) @ zero_zero_real ) ) ) ).

% VEBT_internal.cnt.simps(2)
thf(fact_5204_p1mod22k,axiom,
    ! [B3: int,N2: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ one_one_int ) ) ).

% p1mod22k
thf(fact_5205_p1mod22k_H,axiom,
    ! [B3: int,N2: nat] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% p1mod22k'
thf(fact_5206_VEBT__internal_Ocnt_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2 != one_one_real ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
               != ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList3 ) @ zero_zero_real ) ) ) ) ) ) ).

% VEBT_internal.cnt.elims
thf(fact_5207_pos__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ A3 ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_5208_sb__inc__lem,axiom,
    ! [A3: int,K: nat] :
      ( ( ord_less_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem
thf(fact_5209_neg__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A3 ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_5210_mult__less__iff1,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y2 @ Z ) )
        = ( ord_less_real @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_5211_mult__less__iff1,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y2 @ Z ) )
        = ( ord_less_rat @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_5212_mult__less__iff1,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y2 @ Z ) )
        = ( ord_less_int @ X2 @ Y2 ) ) ) ).

% mult_less_iff1
thf(fact_5213_of__int__round__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_5214_of__int__round__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_5215_of__int__round__ge,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_5216_of__int__round__ge,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_ge
thf(fact_5217_of__int__round__gt,axiom,
    ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_5218_of__int__round__gt,axiom,
    ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).

% of_int_round_gt
thf(fact_5219_mult__le__cancel__iff2,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y2 ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5220_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y2 ) )
        = ( ord_less_eq_rat @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5221_mult__le__cancel__iff2,axiom,
    ! [Z: int,X2: int,Y2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y2 ) )
        = ( ord_less_eq_int @ X2 @ Y2 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_5222_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T3174267988120873389le_nat
    = ( ^ [A_of_integer: code_integer > product_prod_nat_nat,Zero: nat,I4: code_integer] :
          ( if_nat @ ( ord_le6747313008572928689nteger @ I4 @ zero_z3403309356797280102nteger ) @ undefined_nat
          @ ( if_nat @ ( I4 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_nat
              @ ( ( modulo364778990260209775nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( product_snd_nat_nat @ ( A_of_integer @ ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( product_fst_nat_nat @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5223_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ( code_T3171777517611823113le_int
    = ( ^ [A_of_integer: code_integer > product_prod_int_int,Zero: int,I4: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ I4 @ zero_z3403309356797280102nteger ) @ undefined_int
          @ ( if_int @ ( I4 = zero_z3403309356797280102nteger ) @ Zero
            @ ( if_int
              @ ( ( modulo364778990260209775nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = zero_z3403309356797280102nteger )
              @ ( product_snd_int_int @ ( A_of_integer @ ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) )
              @ ( product_fst_int_int @ ( A_of_integer @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
thf(fact_5224_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_5225_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_5226_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2 != one_one_int ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2 != one_one_int ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
thf(fact_5227_refines__assert_H__bind,axiom,
    ! [P2: heap_T2636463487746394924on_nat,Q2: heap_T2636463487746394924on_nat,Phi: $o] :
      ( ( refine7594492741263601813on_nat @ P2 @ Q2 )
     => ( refine7594492741263601813on_nat @ P2
        @ ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5228_refines__assert_H__bind,axiom,
    ! [P2: heap_Time_Heap_o,Q2: heap_Time_Heap_o,Phi: $o] :
      ( ( refine_Imp_refines_o @ P2 @ Q2 )
     => ( refine_Imp_refines_o @ P2
        @ ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5229_refines__assert_H__bind,axiom,
    ! [P2: heap_T8145700208782473153_VEBTi,Q2: heap_T8145700208782473153_VEBTi,Phi: $o] :
      ( ( refine5565527176597971370_VEBTi @ P2 @ Q2 )
     => ( refine5565527176597971370_VEBTi @ P2
        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ Phi )
          @ ^ [Uu: product_unit] : Q2 ) ) ) ).

% refines_assert'_bind
thf(fact_5230_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_5231_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5232_dvd__0__right,axiom,
    ! [A3: uint32] : ( dvd_dvd_uint32 @ A3 @ zero_zero_uint32 ) ).

% dvd_0_right
thf(fact_5233_dvd__0__right,axiom,
    ! [A3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_5234_dvd__0__right,axiom,
    ! [A3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A3 @ zero_z3563351764282998399l_num1 ) ).

% dvd_0_right
thf(fact_5235_dvd__0__right,axiom,
    ! [A3: real] : ( dvd_dvd_real @ A3 @ zero_zero_real ) ).

% dvd_0_right
thf(fact_5236_dvd__0__right,axiom,
    ! [A3: rat] : ( dvd_dvd_rat @ A3 @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_5237_dvd__0__right,axiom,
    ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_5238_dvd__0__right,axiom,
    ! [A3: int] : ( dvd_dvd_int @ A3 @ zero_zero_int ) ).

% dvd_0_right
thf(fact_5239_dvd__0__left__iff,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A3 )
      = ( A3 = zero_zero_uint32 ) ) ).

% dvd_0_left_iff
thf(fact_5240_dvd__0__left__iff,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A3 )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_5241_dvd__0__left__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = ( A3 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left_iff
thf(fact_5242_dvd__0__left__iff,axiom,
    ! [A3: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A3 )
      = ( A3 = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_5243_dvd__0__left__iff,axiom,
    ! [A3: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A3 )
      = ( A3 = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_5244_dvd__0__left__iff,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
      = ( A3 = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_5245_dvd__0__left__iff,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A3 )
      = ( A3 = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_5246_dvd__add__triv__left__iff,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ A3 @ B3 ) )
      = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5247_dvd__add__triv__left__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5248_dvd__add__triv__left__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
      = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5249_dvd__add__triv__left__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ A3 @ B3 ) )
      = ( dvd_dvd_real @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5250_dvd__add__triv__left__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ A3 @ B3 ) )
      = ( dvd_dvd_rat @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5251_dvd__add__triv__left__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( dvd_dvd_nat @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5252_dvd__add__triv__left__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ A3 @ B3 ) )
      = ( dvd_dvd_int @ A3 @ B3 ) ) ).

% dvd_add_triv_left_iff
thf(fact_5253_dvd__add__triv__right__iff,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ B3 @ A3 ) )
      = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5254_dvd__add__triv__right__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ B3 @ A3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5255_dvd__add__triv__right__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ A3 ) )
      = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5256_dvd__add__triv__right__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ A3 ) )
      = ( dvd_dvd_real @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5257_dvd__add__triv__right__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ A3 ) )
      = ( dvd_dvd_rat @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5258_dvd__add__triv__right__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ A3 ) )
      = ( dvd_dvd_nat @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5259_dvd__add__triv__right__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ A3 ) )
      = ( dvd_dvd_int @ A3 @ B3 ) ) ).

% dvd_add_triv_right_iff
thf(fact_5260_div__dvd__div,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ A3 @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ ( divide6298287555418463151nteger @ C @ A3 ) )
          = ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_5261_div__dvd__div,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ A3 @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B3 @ A3 ) @ ( divide_divide_nat @ C @ A3 ) )
          = ( dvd_dvd_nat @ B3 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_5262_div__dvd__div,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ A3 @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B3 @ A3 ) @ ( divide_divide_int @ C @ A3 ) )
          = ( dvd_dvd_int @ B3 @ C ) ) ) ) ).

% div_dvd_div
thf(fact_5263_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_5264_dvd__times__right__cancel__iff,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ ( times_3573771949741848930nteger @ C @ A3 ) )
        = ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5265_dvd__times__right__cancel__iff,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B3 @ A3 ) @ ( times_times_nat @ C @ A3 ) )
        = ( dvd_dvd_nat @ B3 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5266_dvd__times__right__cancel__iff,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( A3 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B3 @ A3 ) @ ( times_times_int @ C @ A3 ) )
        = ( dvd_dvd_int @ B3 @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5267_dvd__times__left__cancel__iff,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ ( times_3573771949741848930nteger @ A3 @ C ) )
        = ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5268_dvd__times__left__cancel__iff,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ ( times_times_nat @ A3 @ C ) )
        = ( dvd_dvd_nat @ B3 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5269_dvd__times__left__cancel__iff,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( A3 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ A3 @ C ) )
        = ( dvd_dvd_int @ B3 @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5270_dvd__mult__cancel__right,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5271_dvd__mult__cancel__right,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5272_dvd__mult__cancel__right,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5273_dvd__mult__cancel__right,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5274_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A3 ) @ ( times_3573771949741848930nteger @ C @ B3 ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5275_dvd__mult__cancel__left,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5276_dvd__mult__cancel__left,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A3 ) @ ( times_times_rat @ C @ B3 ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5277_dvd__mult__cancel__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A3 ) @ ( times_times_int @ C @ B3 ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A3 @ B3 ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5278_dvd__add__times__triv__left__iff,axiom,
    ! [A3: uint32,C: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ ( times_times_uint32 @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5279_dvd__add__times__triv__left__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ C @ A3 ) @ B3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5280_dvd__add__times__triv__left__iff,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5281_dvd__add__times__triv__left__iff,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ ( times_times_real @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_real @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5282_dvd__add__times__triv__left__iff,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ ( times_times_rat @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_rat @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5283_dvd__add__times__triv__left__iff,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ ( times_times_nat @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_nat @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5284_dvd__add__times__triv__left__iff,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ ( times_times_int @ C @ A3 ) @ B3 ) )
      = ( dvd_dvd_int @ A3 @ B3 ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5285_dvd__add__times__triv__right__iff,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ B3 @ ( times_times_uint32 @ C @ A3 ) ) )
      = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5286_dvd__add__times__triv__right__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ B3 @ ( times_7065122842183080059l_num1 @ C @ A3 ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5287_dvd__add__times__triv__right__iff,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ ( times_3573771949741848930nteger @ C @ A3 ) ) )
      = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5288_dvd__add__times__triv__right__iff,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ ( times_times_real @ C @ A3 ) ) )
      = ( dvd_dvd_real @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5289_dvd__add__times__triv__right__iff,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ ( times_times_rat @ C @ A3 ) ) )
      = ( dvd_dvd_rat @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5290_dvd__add__times__triv__right__iff,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ ( times_times_nat @ C @ A3 ) ) )
      = ( dvd_dvd_nat @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5291_dvd__add__times__triv__right__iff,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ ( times_times_int @ C @ A3 ) ) )
      = ( dvd_dvd_int @ A3 @ B3 ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5292_unit__prod,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_5293_unit__prod,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_5294_unit__prod,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( dvd_dvd_int @ B3 @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_5295_div__add,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A3 )
     => ( ( dvd_dvd_Code_integer @ C @ B3 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ) ).

% div_add
thf(fact_5296_div__add,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ C @ A3 )
     => ( ( dvd_dvd_nat @ C @ B3 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ) ).

% div_add
thf(fact_5297_div__add,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ A3 )
     => ( ( dvd_dvd_int @ C @ B3 )
       => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ) ).

% div_add
thf(fact_5298_unit__div,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_5299_unit__div,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_5300_unit__div,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( dvd_dvd_int @ B3 @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_5301_unit__div__1__unit,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_5302_unit__div__1__unit,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_5303_unit__div__1__unit,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A3 ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_5304_unit__div__1__div__1,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) )
        = A3 ) ) ).

% unit_div_1_div_1
thf(fact_5305_unit__div__1__div__1,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A3 ) )
        = A3 ) ) ).

% unit_div_1_div_1
thf(fact_5306_unit__div__1__div__1,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A3 ) )
        = A3 ) ) ).

% unit_div_1_div_1
thf(fact_5307_dvd__mult__div__cancel,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ A3 ) )
        = B3 ) ) ).

% dvd_mult_div_cancel
thf(fact_5308_dvd__mult__div__cancel,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% dvd_mult_div_cancel
thf(fact_5309_dvd__mult__div__cancel,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ A3 ) )
        = B3 ) ) ).

% dvd_mult_div_cancel
thf(fact_5310_dvd__div__mult__self,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% dvd_div_mult_self
thf(fact_5311_dvd__div__mult__self,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% dvd_div_mult_self
thf(fact_5312_dvd__div__mult__self,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( times_times_int @ ( divide_divide_int @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% dvd_div_mult_self
thf(fact_5313_div__diff,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A3 )
     => ( ( dvd_dvd_Code_integer @ C @ B3 )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A3 @ B3 ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ) ).

% div_diff
thf(fact_5314_div__diff,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ A3 )
     => ( ( dvd_dvd_int @ C @ B3 )
       => ( ( divide_divide_int @ ( minus_minus_int @ A3 @ B3 ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ) ).

% div_diff
thf(fact_5315_dvd__imp__mod__0,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( modulo_modulo_nat @ B3 @ A3 )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_5316_dvd__imp__mod__0,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( modulo_modulo_int @ B3 @ A3 )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_5317_dvd__imp__mod__0,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( modulo364778990260209775nteger @ B3 @ A3 )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_5318_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_5319_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_5320_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_5321_unit__mult__div__div,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B3 @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 ) )
        = ( divide6298287555418463151nteger @ B3 @ A3 ) ) ) ).

% unit_mult_div_div
thf(fact_5322_unit__mult__div__div,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( times_times_nat @ B3 @ ( divide_divide_nat @ one_one_nat @ A3 ) )
        = ( divide_divide_nat @ B3 @ A3 ) ) ) ).

% unit_mult_div_div
thf(fact_5323_unit__mult__div__div,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( times_times_int @ B3 @ ( divide_divide_int @ one_one_int @ A3 ) )
        = ( divide_divide_int @ B3 @ A3 ) ) ) ).

% unit_mult_div_div
thf(fact_5324_unit__div__mult__self,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% unit_div_mult_self
thf(fact_5325_unit__div__mult__self,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% unit_div_mult_self
thf(fact_5326_unit__div__mult__self,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B3 @ A3 ) @ A3 )
        = B3 ) ) ).

% unit_div_mult_self
thf(fact_5327_odd__add,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A3 @ B3 ) ) )
      = ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) )
       != ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% odd_add
thf(fact_5328_odd__add,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% odd_add
thf(fact_5329_odd__add,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) ) )
      = ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) )
       != ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% odd_add
thf(fact_5330_odd__add,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ B3 ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% odd_add
thf(fact_5331_odd__add,axiom,
    ! [A3: int,B3: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ B3 ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).

% odd_add
thf(fact_5332_even__add,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A3 @ B3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
        = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_add
thf(fact_5333_even__add,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_add
thf(fact_5334_even__add,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
        = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_add
thf(fact_5335_even__add,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ B3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_add
thf(fact_5336_even__add,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ B3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_add
thf(fact_5337_even__mult__iff,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( times_times_uint32 @ A3 @ B3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
        | ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_mult_iff
thf(fact_5338_even__mult__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_mult_iff
thf(fact_5339_even__mult__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( times_7065122842183080059l_num1 @ A3 @ B3 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
        | ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_mult_iff
thf(fact_5340_even__mult__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A3 @ B3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_mult_iff
thf(fact_5341_even__mult__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A3 @ B3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).

% even_mult_iff
thf(fact_5342_even__mod__2__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_mod_2_iff
thf(fact_5343_even__mod__2__iff,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_mod_2_iff
thf(fact_5344_even__mod__2__iff,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_mod_2_iff
thf(fact_5345_even__mod__2__iff,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_mod_2_iff
thf(fact_5346_even__mod__2__iff,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_mod_2_iff
thf(fact_5347_even__Suc,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% even_Suc
thf(fact_5348_even__Suc__Suc__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_Suc_Suc_iff
thf(fact_5349_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5350_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5351_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5352_even__plus__one__iff,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ).

% even_plus_one_iff
thf(fact_5353_even__plus__one__iff,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A3 @ one_one_uint32 ) )
      = ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) ) ) ).

% even_plus_one_iff
thf(fact_5354_even__plus__one__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A3 @ one_on7727431528512463931l_num1 ) )
      = ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) ) ) ).

% even_plus_one_iff
thf(fact_5355_even__plus__one__iff,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ one_one_nat ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ).

% even_plus_one_iff
thf(fact_5356_even__plus__one__iff,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ one_one_int ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ).

% even_plus_one_iff
thf(fact_5357_even__diff,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ A3 @ B3 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A3 @ B3 ) ) ) ).

% even_diff
thf(fact_5358_even__diff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A3 @ B3 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) ) ).

% even_diff
thf(fact_5359_even__diff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_4019991460397169231l_num1 @ A3 @ B3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) ) ) ).

% even_diff
thf(fact_5360_even__diff,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A3 @ B3 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ B3 ) ) ) ).

% even_diff
thf(fact_5361_odd__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% odd_Suc_div_two
thf(fact_5362_even__Suc__div__two,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_Suc_div_two
thf(fact_5363_odd__succ__div__two,axiom,
    ! [A3: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_5364_odd__succ__div__two,axiom,
    ! [A3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).

% odd_succ_div_two
thf(fact_5365_odd__succ__div__two,axiom,
    ! [A3: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% odd_succ_div_two
thf(fact_5366_even__succ__div__two,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_5367_even__succ__div__two,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_5368_even__succ__div__two,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_5369_even__succ__div__2,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5370_even__succ__div__2,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A3 ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5371_even__succ__div__2,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A3 ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5372_even__succ__div__2,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5373_even__succ__div__2,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_5374_even__power,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( power_power_uint32 @ A3 @ N2 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_5375_even__power,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_5376_even__power,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( power_2184487114949457152l_num1 @ A3 @ N2 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_5377_even__power,axiom,
    ! [A3: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A3 @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_5378_even__power,axiom,
    ! [A3: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A3 @ N2 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_5379_zero__le__power__eq__numeral,axiom,
    ! [A3: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ ( 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 @ A3 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5380_zero__le__power__eq__numeral,axiom,
    ! [A3: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( 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_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5381_zero__le__power__eq__numeral,axiom,
    ! [A3: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( 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 @ A3 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5382_zero__le__power__eq__numeral,axiom,
    ! [A3: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ ( 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 @ A3 ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5383_power__less__zero__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq
thf(fact_5384_power__less__zero__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ ( power_power_real @ A3 @ N2 ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).

% power_less_zero_eq
thf(fact_5385_power__less__zero__eq,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A3 @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_5386_power__less__zero__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ ( power_power_int @ A3 @ N2 ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).

% power_less_zero_eq
thf(fact_5387_power__less__zero__eq__numeral,axiom,
    ! [A3: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5388_power__less__zero__eq__numeral,axiom,
    ! [A3: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A3 @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5389_power__less__zero__eq__numeral,axiom,
    ! [A3: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5390_power__less__zero__eq__numeral,axiom,
    ! [A3: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A3 @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5391_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( semiri2565882477558803405uint32 @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_5392_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( semiri8819519690708144855l_num1 @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_5393_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_5394_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_5395_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_5396_odd__Suc__minus__one,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% odd_Suc_minus_one
thf(fact_5397_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_5398_odd__two__times__div__two__succ,axiom,
    ! [A3: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A3 ) ) ).

% odd_two_times_div_two_succ
thf(fact_5399_odd__two__times__div__two__succ,axiom,
    ! [A3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
        = A3 ) ) ).

% odd_two_times_div_two_succ
thf(fact_5400_odd__two__times__div__two__succ,axiom,
    ! [A3: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
        = A3 ) ) ).

% odd_two_times_div_two_succ
thf(fact_5401_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) @ one_one_uint32 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5402_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5403_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) @ one_on7727431528512463931l_num1 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5404_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5405_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_5406_zero__less__power__eq__numeral,axiom,
    ! [A3: code_integer,W: num] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A3 != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5407_zero__less__power__eq__numeral,axiom,
    ! [A3: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A3 != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5408_zero__less__power__eq__numeral,axiom,
    ! [A3: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A3 != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5409_zero__less__power__eq__numeral,axiom,
    ! [A3: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A3 != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A3 ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5410_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_5411_power__le__zero__eq__numeral,axiom,
    ! [A3: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A3 @ ( 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 @ A3 @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A3 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5412_power__le__zero__eq__numeral,axiom,
    ! [A3: code_integer,W: num] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ W ) ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5413_power__le__zero__eq__numeral,axiom,
    ! [A3: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ ( 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 @ A3 @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A3 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5414_power__le__zero__eq__numeral,axiom,
    ! [A3: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A3 @ ( 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 @ A3 @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A3 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5415_even__succ__div__exp,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5416_even__succ__div__exp,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide1791077408188789448l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A3 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide1791077408188789448l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5417_even__succ__div__exp,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A3 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5418_even__succ__div__exp,axiom,
    ! [A3: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5419_even__succ__div__exp,axiom,
    ! [A3: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_5420_even__succ__mod__exp,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo1504961113040953224l_num1 @ ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ A3 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5421_even__succ__mod__exp,axiom,
    ! [A3: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5422_even__succ__mod__exp,axiom,
    ! [A3: int,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5423_even__succ__mod__exp,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A3 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5424_even__succ__mod__exp,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo_modulo_uint32 @ ( plus_plus_uint32 @ one_one_uint32 @ A3 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_plus_uint32 @ one_one_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5425_dvd__refl,axiom,
    ! [A3: nat] : ( dvd_dvd_nat @ A3 @ A3 ) ).

% dvd_refl
thf(fact_5426_dvd__refl,axiom,
    ! [A3: int] : ( dvd_dvd_int @ A3 @ A3 ) ).

% dvd_refl
thf(fact_5427_dvd__refl,axiom,
    ! [A3: uint32] : ( dvd_dvd_uint32 @ A3 @ A3 ) ).

% dvd_refl
thf(fact_5428_dvd__refl,axiom,
    ! [A3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A3 @ A3 ) ).

% dvd_refl
thf(fact_5429_dvd__refl,axiom,
    ! [A3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ A3 ) ).

% dvd_refl
thf(fact_5430_dvd__trans,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ B3 @ C )
       => ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% dvd_trans
thf(fact_5431_dvd__trans,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ B3 @ C )
       => ( dvd_dvd_int @ A3 @ C ) ) ) ).

% dvd_trans
thf(fact_5432_dvd__trans,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ B3 )
     => ( ( dvd_dvd_uint32 @ B3 @ C )
       => ( dvd_dvd_uint32 @ A3 @ C ) ) ) ).

% dvd_trans
thf(fact_5433_dvd__trans,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ B3 @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A3 @ C ) ) ) ).

% dvd_trans
thf(fact_5434_dvd__trans,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ B3 @ C )
       => ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% dvd_trans
thf(fact_5435_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_5436_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5437_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5438_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5439_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A2: real,B2: real] :
          ( ( A2 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_5440_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A2: rat,B2: rat] :
          ( ( A2 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_5441_dvd__0__left,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ zero_zero_uint32 @ A3 )
     => ( A3 = zero_zero_uint32 ) ) ).

% dvd_0_left
thf(fact_5442_dvd__0__left,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A3 )
     => ( A3 = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_5443_dvd__0__left,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
     => ( A3 = zero_z3563351764282998399l_num1 ) ) ).

% dvd_0_left
thf(fact_5444_dvd__0__left,axiom,
    ! [A3: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A3 )
     => ( A3 = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_5445_dvd__0__left,axiom,
    ! [A3: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A3 )
     => ( A3 = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_5446_dvd__0__left,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
     => ( A3 = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_5447_dvd__0__left,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A3 )
     => ( A3 = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_5448_dvd__add,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ B3 )
     => ( ( dvd_dvd_uint32 @ A3 @ C )
       => ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5449_dvd__add,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ A3 @ C )
       => ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5450_dvd__add,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ A3 @ C )
       => ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5451_dvd__add,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ A3 @ B3 )
     => ( ( dvd_dvd_real @ A3 @ C )
       => ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5452_dvd__add,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ A3 @ B3 )
     => ( ( dvd_dvd_rat @ A3 @ C )
       => ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5453_dvd__add,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ A3 @ C )
       => ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5454_dvd__add,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ A3 @ C )
       => ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) ) ) ) ).

% dvd_add
thf(fact_5455_dvd__add__left__iff,axiom,
    ! [A3: uint32,C: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ C )
     => ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ B3 @ C ) )
        = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5456_dvd__add__left__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ C )
     => ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ B3 @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5457_dvd__add__left__iff,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ C )
     => ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) )
        = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5458_dvd__add__left__iff,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( dvd_dvd_real @ A3 @ C )
     => ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
        = ( dvd_dvd_real @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5459_dvd__add__left__iff,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( dvd_dvd_rat @ A3 @ C )
     => ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
        = ( dvd_dvd_rat @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5460_dvd__add__left__iff,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ C )
     => ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
        = ( dvd_dvd_nat @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5461_dvd__add__left__iff,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ C )
     => ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
        = ( dvd_dvd_int @ A3 @ B3 ) ) ) ).

% dvd_add_left_iff
thf(fact_5462_dvd__add__right__iff,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ B3 )
     => ( ( dvd_dvd_uint32 @ A3 @ ( plus_plus_uint32 @ B3 @ C ) )
        = ( dvd_dvd_uint32 @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5463_dvd__add__right__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ A3 @ ( plus_p361126936061061375l_num1 @ B3 @ C ) )
        = ( dvd_dv6812691276156420380l_num1 @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5464_dvd__add__right__iff,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ A3 @ ( plus_p5714425477246183910nteger @ B3 @ C ) )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5465_dvd__add__right__iff,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ A3 @ B3 )
     => ( ( dvd_dvd_real @ A3 @ ( plus_plus_real @ B3 @ C ) )
        = ( dvd_dvd_real @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5466_dvd__add__right__iff,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ A3 @ B3 )
     => ( ( dvd_dvd_rat @ A3 @ ( plus_plus_rat @ B3 @ C ) )
        = ( dvd_dvd_rat @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5467_dvd__add__right__iff,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ A3 @ ( plus_plus_nat @ B3 @ C ) )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5468_dvd__add__right__iff,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ A3 @ ( plus_plus_int @ B3 @ C ) )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5469_one__dvd,axiom,
    ! [A3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ one_on7727431528512463931l_num1 @ A3 ) ).

% one_dvd
thf(fact_5470_one__dvd,axiom,
    ! [A3: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A3 ) ).

% one_dvd
thf(fact_5471_one__dvd,axiom,
    ! [A3: uint32] : ( dvd_dvd_uint32 @ one_one_uint32 @ A3 ) ).

% one_dvd
thf(fact_5472_one__dvd,axiom,
    ! [A3: real] : ( dvd_dvd_real @ one_one_real @ A3 ) ).

% one_dvd
thf(fact_5473_one__dvd,axiom,
    ! [A3: rat] : ( dvd_dvd_rat @ one_one_rat @ A3 ) ).

% one_dvd
thf(fact_5474_one__dvd,axiom,
    ! [A3: nat] : ( dvd_dvd_nat @ one_one_nat @ A3 ) ).

% one_dvd
thf(fact_5475_one__dvd,axiom,
    ! [A3: int] : ( dvd_dvd_int @ one_one_int @ A3 ) ).

% one_dvd
thf(fact_5476_unit__imp__dvd,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).

% unit_imp_dvd
thf(fact_5477_unit__imp__dvd,axiom,
    ! [B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( dvd_dvd_nat @ B3 @ A3 ) ) ).

% unit_imp_dvd
thf(fact_5478_unit__imp__dvd,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( dvd_dvd_int @ B3 @ A3 ) ) ).

% unit_imp_dvd
thf(fact_5479_dvd__unit__imp__unit,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5480_dvd__unit__imp__unit,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
       => ( dvd_dvd_nat @ A3 @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5481_dvd__unit__imp__unit,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ B3 @ one_one_int )
       => ( dvd_dvd_int @ A3 @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5482_dvdE,axiom,
    ! [B3: uint32,A3: uint32] :
      ( ( dvd_dvd_uint32 @ B3 @ A3 )
     => ~ ! [K2: uint32] :
            ( A3
           != ( times_times_uint32 @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5483_dvdE,axiom,
    ! [B3: word_N3645301735248828278l_num1,A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ B3 @ A3 )
     => ~ ! [K2: word_N3645301735248828278l_num1] :
            ( A3
           != ( times_7065122842183080059l_num1 @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5484_dvdE,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ~ ! [K2: code_integer] :
            ( A3
           != ( times_3573771949741848930nteger @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5485_dvdE,axiom,
    ! [B3: real,A3: real] :
      ( ( dvd_dvd_real @ B3 @ A3 )
     => ~ ! [K2: real] :
            ( A3
           != ( times_times_real @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5486_dvdE,axiom,
    ! [B3: rat,A3: rat] :
      ( ( dvd_dvd_rat @ B3 @ A3 )
     => ~ ! [K2: rat] :
            ( A3
           != ( times_times_rat @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5487_dvdE,axiom,
    ! [B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ A3 )
     => ~ ! [K2: nat] :
            ( A3
           != ( times_times_nat @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5488_dvdE,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ~ ! [K2: int] :
            ( A3
           != ( times_times_int @ B3 @ K2 ) ) ) ).

% dvdE
thf(fact_5489_dvdI,axiom,
    ! [A3: uint32,B3: uint32,K: uint32] :
      ( ( A3
        = ( times_times_uint32 @ B3 @ K ) )
     => ( dvd_dvd_uint32 @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5490_dvdI,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,K: word_N3645301735248828278l_num1] :
      ( ( A3
        = ( times_7065122842183080059l_num1 @ B3 @ K ) )
     => ( dvd_dv6812691276156420380l_num1 @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5491_dvdI,axiom,
    ! [A3: code_integer,B3: code_integer,K: code_integer] :
      ( ( A3
        = ( times_3573771949741848930nteger @ B3 @ K ) )
     => ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5492_dvdI,axiom,
    ! [A3: real,B3: real,K: real] :
      ( ( A3
        = ( times_times_real @ B3 @ K ) )
     => ( dvd_dvd_real @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5493_dvdI,axiom,
    ! [A3: rat,B3: rat,K: rat] :
      ( ( A3
        = ( times_times_rat @ B3 @ K ) )
     => ( dvd_dvd_rat @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5494_dvdI,axiom,
    ! [A3: nat,B3: nat,K: nat] :
      ( ( A3
        = ( times_times_nat @ B3 @ K ) )
     => ( dvd_dvd_nat @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5495_dvdI,axiom,
    ! [A3: int,B3: int,K: int] :
      ( ( A3
        = ( times_times_int @ B3 @ K ) )
     => ( dvd_dvd_int @ B3 @ A3 ) ) ).

% dvdI
thf(fact_5496_dvd__def,axiom,
    ( dvd_dvd_uint32
    = ( ^ [B2: uint32,A2: uint32] :
        ? [K3: uint32] :
          ( A2
          = ( times_times_uint32 @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5497_dvd__def,axiom,
    ( dvd_dv6812691276156420380l_num1
    = ( ^ [B2: word_N3645301735248828278l_num1,A2: word_N3645301735248828278l_num1] :
        ? [K3: word_N3645301735248828278l_num1] :
          ( A2
          = ( times_7065122842183080059l_num1 @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5498_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A2: code_integer] :
        ? [K3: code_integer] :
          ( A2
          = ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5499_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A2: real] :
        ? [K3: real] :
          ( A2
          = ( times_times_real @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5500_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B2: rat,A2: rat] :
        ? [K3: rat] :
          ( A2
          = ( times_times_rat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5501_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A2: nat] :
        ? [K3: nat] :
          ( A2
          = ( times_times_nat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5502_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A2: int] :
        ? [K3: int] :
          ( A2
          = ( times_times_int @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5503_dvd__mult,axiom,
    ! [A3: uint32,C: uint32,B3: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ C )
     => ( dvd_dvd_uint32 @ A3 @ ( times_times_uint32 @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5504_dvd__mult,axiom,
    ! [A3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A3 @ ( times_7065122842183080059l_num1 @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5505_dvd__mult,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ C )
     => ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5506_dvd__mult,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( dvd_dvd_real @ A3 @ C )
     => ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5507_dvd__mult,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( dvd_dvd_rat @ A3 @ C )
     => ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5508_dvd__mult,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ C )
     => ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5509_dvd__mult,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ C )
     => ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).

% dvd_mult
thf(fact_5510_dvd__mult2,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ B3 )
     => ( dvd_dvd_uint32 @ A3 @ ( times_times_uint32 @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5511_dvd__mult2,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
     => ( dvd_dv6812691276156420380l_num1 @ A3 @ ( times_7065122842183080059l_num1 @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5512_dvd__mult2,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5513_dvd__mult2,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ A3 @ B3 )
     => ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5514_dvd__mult2,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ A3 @ B3 )
     => ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5515_dvd__mult2,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5516_dvd__mult2,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) ) ) ).

% dvd_mult2
thf(fact_5517_dvd__mult__left,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A3 @ B3 ) @ C )
     => ( dvd_dvd_uint32 @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5518_dvd__mult__left,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ B3 ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5519_dvd__mult__left,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
     => ( dvd_dvd_Code_integer @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5520_dvd__mult__left,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B3 ) @ C )
     => ( dvd_dvd_real @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5521_dvd__mult__left,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
     => ( dvd_dvd_rat @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5522_dvd__mult__left,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
     => ( dvd_dvd_nat @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5523_dvd__mult__left,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
     => ( dvd_dvd_int @ A3 @ C ) ) ).

% dvd_mult_left
thf(fact_5524_dvd__triv__left,axiom,
    ! [A3: uint32,B3: uint32] : ( dvd_dvd_uint32 @ A3 @ ( times_times_uint32 @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5525_dvd__triv__left,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A3 @ ( times_7065122842183080059l_num1 @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5526_dvd__triv__left,axiom,
    ! [A3: code_integer,B3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5527_dvd__triv__left,axiom,
    ! [A3: real,B3: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5528_dvd__triv__left,axiom,
    ! [A3: rat,B3: rat] : ( dvd_dvd_rat @ A3 @ ( times_times_rat @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5529_dvd__triv__left,axiom,
    ! [A3: nat,B3: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5530_dvd__triv__left,axiom,
    ! [A3: int,B3: int] : ( dvd_dvd_int @ A3 @ ( times_times_int @ A3 @ B3 ) ) ).

% dvd_triv_left
thf(fact_5531_mult__dvd__mono,axiom,
    ! [A3: uint32,B3: uint32,C: uint32,D: uint32] :
      ( ( dvd_dvd_uint32 @ A3 @ B3 )
     => ( ( dvd_dvd_uint32 @ C @ D )
       => ( dvd_dvd_uint32 @ ( times_times_uint32 @ A3 @ C ) @ ( times_times_uint32 @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5532_mult__dvd__mono,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1,D: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
     => ( ( dvd_dv6812691276156420380l_num1 @ C @ D )
       => ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ C ) @ ( times_7065122842183080059l_num1 @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5533_mult__dvd__mono,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ B3 )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5534_mult__dvd__mono,axiom,
    ! [A3: real,B3: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A3 @ B3 )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5535_mult__dvd__mono,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A3 @ B3 )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A3 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5536_mult__dvd__mono,axiom,
    ! [A3: nat,B3: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A3 @ B3 )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5537_mult__dvd__mono,axiom,
    ! [A3: int,B3: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A3 @ B3 )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5538_dvd__mult__right,axiom,
    ! [A3: uint32,B3: uint32,C: uint32] :
      ( ( dvd_dvd_uint32 @ ( times_times_uint32 @ A3 @ B3 ) @ C )
     => ( dvd_dvd_uint32 @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5539_dvd__mult__right,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1,C: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ B3 ) @ C )
     => ( dvd_dv6812691276156420380l_num1 @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5540_dvd__mult__right,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
     => ( dvd_dvd_Code_integer @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5541_dvd__mult__right,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A3 @ B3 ) @ C )
     => ( dvd_dvd_real @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5542_dvd__mult__right,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A3 @ B3 ) @ C )
     => ( dvd_dvd_rat @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5543_dvd__mult__right,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
     => ( dvd_dvd_nat @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5544_dvd__mult__right,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
     => ( dvd_dvd_int @ B3 @ C ) ) ).

% dvd_mult_right
thf(fact_5545_dvd__triv__right,axiom,
    ! [A3: uint32,B3: uint32] : ( dvd_dvd_uint32 @ A3 @ ( times_times_uint32 @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5546_dvd__triv__right,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ A3 @ ( times_7065122842183080059l_num1 @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5547_dvd__triv__right,axiom,
    ! [A3: code_integer,B3: code_integer] : ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5548_dvd__triv__right,axiom,
    ! [A3: real,B3: real] : ( dvd_dvd_real @ A3 @ ( times_times_real @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5549_dvd__triv__right,axiom,
    ! [A3: rat,B3: rat] : ( dvd_dvd_rat @ A3 @ ( times_times_rat @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5550_dvd__triv__right,axiom,
    ! [A3: nat,B3: nat] : ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5551_dvd__triv__right,axiom,
    ! [A3: int,B3: int] : ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ A3 ) ) ).

% dvd_triv_right
thf(fact_5552_dvd__diff,axiom,
    ! [X2: uint32,Y2: uint32,Z: uint32] :
      ( ( dvd_dvd_uint32 @ X2 @ Y2 )
     => ( ( dvd_dvd_uint32 @ X2 @ Z )
       => ( dvd_dvd_uint32 @ X2 @ ( minus_minus_uint32 @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5553_dvd__diff,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,Z: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ X2 @ Y2 )
     => ( ( dvd_dv6812691276156420380l_num1 @ X2 @ Z )
       => ( dvd_dv6812691276156420380l_num1 @ X2 @ ( minus_4019991460397169231l_num1 @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5554_dvd__diff,axiom,
    ! [X2: code_integer,Y2: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X2 @ Y2 )
     => ( ( dvd_dvd_Code_integer @ X2 @ Z )
       => ( dvd_dvd_Code_integer @ X2 @ ( minus_8373710615458151222nteger @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5555_dvd__diff,axiom,
    ! [X2: real,Y2: real,Z: real] :
      ( ( dvd_dvd_real @ X2 @ Y2 )
     => ( ( dvd_dvd_real @ X2 @ Z )
       => ( dvd_dvd_real @ X2 @ ( minus_minus_real @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5556_dvd__diff,axiom,
    ! [X2: rat,Y2: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X2 @ Y2 )
     => ( ( dvd_dvd_rat @ X2 @ Z )
       => ( dvd_dvd_rat @ X2 @ ( minus_minus_rat @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5557_dvd__diff,axiom,
    ! [X2: int,Y2: int,Z: int] :
      ( ( dvd_dvd_int @ X2 @ Y2 )
     => ( ( dvd_dvd_int @ X2 @ Z )
       => ( dvd_dvd_int @ X2 @ ( minus_minus_int @ Y2 @ Z ) ) ) ) ).

% dvd_diff
thf(fact_5558_dvd__diff__commute,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ ( minus_8373710615458151222nteger @ C @ B3 ) )
      = ( dvd_dvd_Code_integer @ A3 @ ( minus_8373710615458151222nteger @ B3 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_5559_dvd__diff__commute,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ ( minus_minus_int @ C @ B3 ) )
      = ( dvd_dvd_int @ A3 @ ( minus_minus_int @ B3 @ C ) ) ) ).

% dvd_diff_commute
thf(fact_5560_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A3 )
     => ( ( dvd_dvd_Code_integer @ C @ B3 )
       => ( ( ( divide6298287555418463151nteger @ A3 @ C )
            = ( divide6298287555418463151nteger @ B3 @ C ) )
          = ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5561_dvd__div__eq__iff,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( dvd_dvd_real @ C @ A3 )
     => ( ( dvd_dvd_real @ C @ B3 )
       => ( ( ( divide_divide_real @ A3 @ C )
            = ( divide_divide_real @ B3 @ C ) )
          = ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5562_dvd__div__eq__iff,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( dvd_dvd_rat @ C @ A3 )
     => ( ( dvd_dvd_rat @ C @ B3 )
       => ( ( ( divide_divide_rat @ A3 @ C )
            = ( divide_divide_rat @ B3 @ C ) )
          = ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5563_dvd__div__eq__iff,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ C @ A3 )
     => ( ( dvd_dvd_nat @ C @ B3 )
       => ( ( ( divide_divide_nat @ A3 @ C )
            = ( divide_divide_nat @ B3 @ C ) )
          = ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5564_dvd__div__eq__iff,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ A3 )
     => ( ( dvd_dvd_int @ C @ B3 )
       => ( ( ( divide_divide_int @ A3 @ C )
            = ( divide_divide_int @ B3 @ C ) )
          = ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5565_dvd__div__eq__cancel,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A3 @ C )
        = ( divide6298287555418463151nteger @ B3 @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A3 )
       => ( ( dvd_dvd_Code_integer @ C @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5566_dvd__div__eq__cancel,axiom,
    ! [A3: real,C: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ C )
        = ( divide_divide_real @ B3 @ C ) )
     => ( ( dvd_dvd_real @ C @ A3 )
       => ( ( dvd_dvd_real @ C @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5567_dvd__div__eq__cancel,axiom,
    ! [A3: rat,C: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A3 @ C )
        = ( divide_divide_rat @ B3 @ C ) )
     => ( ( dvd_dvd_rat @ C @ A3 )
       => ( ( dvd_dvd_rat @ C @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5568_dvd__div__eq__cancel,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( ( divide_divide_nat @ A3 @ C )
        = ( divide_divide_nat @ B3 @ C ) )
     => ( ( dvd_dvd_nat @ C @ A3 )
       => ( ( dvd_dvd_nat @ C @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5569_dvd__div__eq__cancel,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ( divide_divide_int @ A3 @ C )
        = ( divide_divide_int @ B3 @ C ) )
     => ( ( dvd_dvd_int @ C @ A3 )
       => ( ( dvd_dvd_int @ C @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5570_div__div__div__same,axiom,
    ! [D: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B3 )
     => ( ( dvd_dvd_Code_integer @ B3 @ A3 )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ D ) @ ( divide6298287555418463151nteger @ B3 @ D ) )
          = ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).

% div_div_div_same
thf(fact_5571_div__div__div__same,axiom,
    ! [D: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ D @ B3 )
     => ( ( dvd_dvd_nat @ B3 @ A3 )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A3 @ D ) @ ( divide_divide_nat @ B3 @ D ) )
          = ( divide_divide_nat @ A3 @ B3 ) ) ) ) ).

% div_div_div_same
thf(fact_5572_div__div__div__same,axiom,
    ! [D: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ D @ B3 )
     => ( ( dvd_dvd_int @ B3 @ A3 )
       => ( ( divide_divide_int @ ( divide_divide_int @ A3 @ D ) @ ( divide_divide_int @ B3 @ D ) )
          = ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% div_div_div_same
thf(fact_5573_dvd__power__same,axiom,
    ! [X2: uint32,Y2: uint32,N2: nat] :
      ( ( dvd_dvd_uint32 @ X2 @ Y2 )
     => ( dvd_dvd_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ ( power_power_uint32 @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5574_dvd__power__same,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ X2 @ Y2 )
     => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ ( power_2184487114949457152l_num1 @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5575_dvd__power__same,axiom,
    ! [X2: nat,Y2: nat,N2: nat] :
      ( ( dvd_dvd_nat @ X2 @ Y2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5576_dvd__power__same,axiom,
    ! [X2: int,Y2: int,N2: nat] :
      ( ( dvd_dvd_int @ X2 @ Y2 )
     => ( dvd_dvd_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5577_dvd__power__same,axiom,
    ! [X2: real,Y2: real,N2: nat] :
      ( ( dvd_dvd_real @ X2 @ Y2 )
     => ( dvd_dvd_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5578_dvd__power__same,axiom,
    ! [X2: complex,Y2: complex,N2: nat] :
      ( ( dvd_dvd_complex @ X2 @ Y2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5579_dvd__power__same,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ X2 @ Y2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ ( power_8256067586552552935nteger @ Y2 @ N2 ) ) ) ).

% dvd_power_same
thf(fact_5580_dvd__mod__iff,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
        = ( dvd_dvd_nat @ C @ A3 ) ) ) ).

% dvd_mod_iff
thf(fact_5581_dvd__mod__iff,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
        = ( dvd_dvd_int @ C @ A3 ) ) ) ).

% dvd_mod_iff
thf(fact_5582_dvd__mod__iff,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
        = ( dvd_dvd_Code_integer @ C @ A3 ) ) ) ).

% dvd_mod_iff
thf(fact_5583_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A3 @ B3 ) )
     => ( ( dvd_dvd_nat @ C @ B3 )
       => ( dvd_dvd_nat @ C @ A3 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5584_dvd__mod__imp__dvd,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A3 @ B3 ) )
     => ( ( dvd_dvd_int @ C @ B3 )
       => ( dvd_dvd_int @ C @ A3 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5585_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A3 @ B3 ) )
     => ( ( dvd_dvd_Code_integer @ C @ B3 )
       => ( dvd_dvd_Code_integer @ C @ A3 ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5586_dvd__mod,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_5587_dvd__mod,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_5588_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N2: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N2 )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_5589_mod__mod__cancel,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A3 @ B3 ) @ C )
        = ( modulo_modulo_nat @ A3 @ C ) ) ) ).

% mod_mod_cancel
thf(fact_5590_mod__mod__cancel,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A3 @ B3 ) @ C )
        = ( modulo_modulo_int @ A3 @ C ) ) ) ).

% mod_mod_cancel
thf(fact_5591_mod__mod__cancel,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) @ C )
        = ( modulo364778990260209775nteger @ A3 @ C ) ) ) ).

% mod_mod_cancel
thf(fact_5592_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_5593_subset__divisors__dvd,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A3 ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B3 ) ) )
      = ( dvd_dvd_complex @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5594_subset__divisors__dvd,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A3 ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B3 ) ) )
      = ( dvd_dvd_int @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5595_subset__divisors__dvd,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ord_le2219237028632753026uint32
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ A3 ) )
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ B3 ) ) )
      = ( dvd_dvd_uint32 @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5596_subset__divisors__dvd,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ord_le5203802739334966412l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ A3 ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ B3 ) ) )
      = ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5597_subset__divisors__dvd,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A3 ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B3 ) ) )
      = ( dvd_dvd_Code_integer @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5598_subset__divisors__dvd,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A3 ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B3 ) ) )
      = ( dvd_dvd_nat @ A3 @ B3 ) ) ).

% subset_divisors_dvd
thf(fact_5599_strict__subset__divisors__dvd,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A3 ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B3 ) ) )
      = ( ( dvd_dvd_complex @ A3 @ B3 )
        & ~ ( dvd_dvd_complex @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5600_strict__subset__divisors__dvd,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A3 ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B3 ) ) )
      = ( ( dvd_dvd_nat @ A3 @ B3 )
        & ~ ( dvd_dvd_nat @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5601_strict__subset__divisors__dvd,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A3 ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B3 ) ) )
      = ( ( dvd_dvd_int @ A3 @ B3 )
        & ~ ( dvd_dvd_int @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5602_strict__subset__divisors__dvd,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ord_less_set_uint32
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ A3 ) )
        @ ( collect_uint32
          @ ^ [C3: uint32] : ( dvd_dvd_uint32 @ C3 @ B3 ) ) )
      = ( ( dvd_dvd_uint32 @ A3 @ B3 )
        & ~ ( dvd_dvd_uint32 @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5603_strict__subset__divisors__dvd,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ord_le6726900395242856064l_num1
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ A3 ) )
        @ ( collec7814023847061821259l_num1
          @ ^ [C3: word_N3645301735248828278l_num1] : ( dvd_dv6812691276156420380l_num1 @ C3 @ B3 ) ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ A3 @ B3 )
        & ~ ( dvd_dv6812691276156420380l_num1 @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5604_strict__subset__divisors__dvd,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A3 ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B3 ) ) )
      = ( ( dvd_dvd_Code_integer @ A3 @ B3 )
        & ~ ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5605_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_5606_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_5607_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_5608_minf_I10_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z4: uint32] :
    ! [X4: uint32] :
      ( ( ord_less_uint32 @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5609_minf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z4: word_N3645301735248828278l_num1] :
    ! [X4: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X4 @ Z4 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5610_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5611_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5612_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5613_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5614_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5615_minf_I9_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z4: uint32] :
    ! [X4: uint32] :
      ( ( ord_less_uint32 @ X4 @ Z4 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5616_minf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z4: word_N3645301735248828278l_num1] :
    ! [X4: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ X4 @ Z4 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5617_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X4 @ Z4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5618_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ X4 @ Z4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5619_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ X4 @ Z4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5620_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5621_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ Z4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% minf(9)
thf(fact_5622_pinf_I10_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z4: uint32] :
    ! [X4: uint32] :
      ( ( ord_less_uint32 @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5623_pinf_I10_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z4: word_N3645301735248828278l_num1] :
    ! [X4: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z4 @ X4 )
     => ( ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) )
        = ( ~ ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5624_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5625_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5626_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5627_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5628_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5629_pinf_I9_J,axiom,
    ! [D: uint32,S: uint32] :
    ? [Z4: uint32] :
    ! [X4: uint32] :
      ( ( ord_less_uint32 @ Z4 @ X4 )
     => ( ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) )
        = ( dvd_dvd_uint32 @ D @ ( plus_plus_uint32 @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5630_pinf_I9_J,axiom,
    ! [D: word_N3645301735248828278l_num1,S: word_N3645301735248828278l_num1] :
    ? [Z4: word_N3645301735248828278l_num1] :
    ! [X4: word_N3645301735248828278l_num1] :
      ( ( ord_le750835935415966154l_num1 @ Z4 @ X4 )
     => ( ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) )
        = ( dvd_dv6812691276156420380l_num1 @ D @ ( plus_p361126936061061375l_num1 @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5631_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z4: code_integer] :
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z4 @ X4 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5632_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z4: real] :
    ! [X4: real] :
      ( ( ord_less_real @ Z4 @ X4 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5633_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z4: rat] :
    ! [X4: rat] :
      ( ( ord_less_rat @ Z4 @ X4 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5634_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z4: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z4 @ X4 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5635_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z4: int] :
    ! [X4: int] :
      ( ( ord_less_int @ Z4 @ X4 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ S ) ) ) ) ).

% pinf(9)
thf(fact_5636_dvd__div__eq__0__iff,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
          = zero_z3403309356797280102nteger )
        = ( A3 = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5637_dvd__div__eq__0__iff,axiom,
    ! [B3: real,A3: real] :
      ( ( dvd_dvd_real @ B3 @ A3 )
     => ( ( ( divide_divide_real @ A3 @ B3 )
          = zero_zero_real )
        = ( A3 = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5638_dvd__div__eq__0__iff,axiom,
    ! [B3: rat,A3: rat] :
      ( ( dvd_dvd_rat @ B3 @ A3 )
     => ( ( ( divide_divide_rat @ A3 @ B3 )
          = zero_zero_rat )
        = ( A3 = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5639_dvd__div__eq__0__iff,axiom,
    ! [B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ A3 )
     => ( ( ( divide_divide_nat @ A3 @ B3 )
          = zero_zero_nat )
        = ( A3 = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5640_dvd__div__eq__0__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ( ( ( divide_divide_int @ A3 @ B3 )
          = zero_zero_int )
        = ( A3 = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5641_unit__mult__right__cancel,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B3 @ A3 )
          = ( times_3573771949741848930nteger @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_5642_unit__mult__right__cancel,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( ( times_times_nat @ B3 @ A3 )
          = ( times_times_nat @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_5643_unit__mult__right__cancel,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( ( times_times_int @ B3 @ A3 )
          = ( times_times_int @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_5644_unit__mult__left__cancel,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A3 @ B3 )
          = ( times_3573771949741848930nteger @ A3 @ C ) )
        = ( B3 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_5645_unit__mult__left__cancel,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( ( times_times_nat @ A3 @ B3 )
          = ( times_times_nat @ A3 @ C ) )
        = ( B3 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_5646_unit__mult__left__cancel,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( ( times_times_int @ A3 @ B3 )
          = ( times_times_int @ A3 @ C ) )
        = ( B3 = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_5647_mult__unit__dvd__iff_H,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
        = ( dvd_dvd_Code_integer @ B3 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5648_mult__unit__dvd__iff_H,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
        = ( dvd_dvd_nat @ B3 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5649_mult__unit__dvd__iff_H,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
        = ( dvd_dvd_int @ B3 @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5650_dvd__mult__unit__iff_H,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5651_dvd__mult__unit__iff_H,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5652_dvd__mult__unit__iff_H,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5653_mult__unit__dvd__iff,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5654_mult__unit__dvd__iff,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ C )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5655_mult__unit__dvd__iff,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ C )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5656_dvd__mult__unit__iff,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5657_dvd__mult__unit__iff,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C @ B3 ) )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5658_dvd__mult__unit__iff,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ A3 @ ( times_times_int @ C @ B3 ) )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5659_is__unit__mult__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_5660_is__unit__mult__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A3 @ one_one_nat )
        & ( dvd_dvd_nat @ B3 @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_5661_is__unit__mult__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A3 @ B3 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A3 @ one_one_int )
        & ( dvd_dvd_int @ B3 @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_5662_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A3 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5663_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ C @ A3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5664_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ A3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5665_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A3 @ C ) @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5666_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A3 @ B3 ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A3 @ C ) @ ( divide_divide_nat @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5667_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( divide_divide_int @ ( plus_plus_int @ A3 @ B3 ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A3 @ C ) @ ( divide_divide_int @ B3 @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5668_unit__div__cancel,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
          = ( divide6298287555418463151nteger @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_div_cancel
thf(fact_5669_unit__div__cancel,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ( ( ( divide_divide_nat @ B3 @ A3 )
          = ( divide_divide_nat @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_div_cancel
thf(fact_5670_unit__div__cancel,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ( ( ( divide_divide_int @ B3 @ A3 )
          = ( divide_divide_int @ C @ A3 ) )
        = ( B3 = C ) ) ) ).

% unit_div_cancel
thf(fact_5671_div__unit__dvd__iff,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_5672_div__unit__dvd__iff,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_5673_div__unit__dvd__iff,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_5674_dvd__div__unit__iff,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ C @ B3 ) )
        = ( dvd_dvd_Code_integer @ A3 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_5675_dvd__div__unit__iff,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ C @ B3 ) )
        = ( dvd_dvd_nat @ A3 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_5676_dvd__div__unit__iff,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ A3 @ ( divide_divide_int @ C @ B3 ) )
        = ( dvd_dvd_int @ A3 @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_5677_dvd__div__mult,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B3 @ C ) @ A3 )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B3 @ A3 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_5678_dvd__div__mult,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( times_times_nat @ ( divide_divide_nat @ B3 @ C ) @ A3 )
        = ( divide_divide_nat @ ( times_times_nat @ B3 @ A3 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_5679_dvd__div__mult,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( times_times_int @ ( divide_divide_int @ B3 @ C ) @ A3 )
        = ( divide_divide_int @ ( times_times_int @ B3 @ A3 ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_5680_div__mult__swap,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_5681_div__mult__swap,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_5682_div__mult__swap,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ) ).

% div_mult_swap
thf(fact_5683_div__div__eq__right,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B3 )
     => ( ( dvd_dvd_Code_integer @ B3 @ A3 )
       => ( ( divide6298287555418463151nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_5684_div__div__eq__right,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ B3 )
     => ( ( dvd_dvd_nat @ B3 @ A3 )
       => ( ( divide_divide_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_5685_div__div__eq__right,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ B3 )
     => ( ( dvd_dvd_int @ B3 @ A3 )
       => ( ( divide_divide_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_5686_dvd__div__mult2__eq,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B3 @ C ) @ A3 )
     => ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_5687_dvd__div__mult2__eq,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B3 @ C ) @ A3 )
     => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_5688_dvd__div__mult2__eq,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B3 @ C ) @ A3 )
     => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_5689_dvd__mult__imp__div,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 )
     => ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_5690_dvd__mult__imp__div,axiom,
    ! [A3: nat,C: nat,B3: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ B3 )
     => ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_5691_dvd__mult__imp__div,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ B3 )
     => ( dvd_dvd_int @ A3 @ ( divide_divide_int @ B3 @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_5692_div__mult__div__if__dvd,axiom,
    ! [B3: code_integer,A3: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ ( times_3573771949741848930nteger @ B3 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_5693_div__mult__div__if__dvd,axiom,
    ! [B3: nat,A3: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ A3 )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_5694_div__mult__div__if__dvd,axiom,
    ! [B3: int,A3: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_5695_div__power,axiom,
    ! [B3: code_integer,A3: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ N2 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ) ).

% div_power
thf(fact_5696_div__power,axiom,
    ! [B3: nat,A3: nat,N2: nat] :
      ( ( dvd_dvd_nat @ B3 @ A3 )
     => ( ( power_power_nat @ ( divide_divide_nat @ A3 @ B3 ) @ N2 )
        = ( divide_divide_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) ) ) ) ).

% div_power
thf(fact_5697_div__power,axiom,
    ! [B3: int,A3: int,N2: nat] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ( ( power_power_int @ ( divide_divide_int @ A3 @ B3 ) @ N2 )
        = ( divide_divide_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ) ).

% div_power
thf(fact_5698_mod__0__imp__dvd,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( modulo1504961113040953224l_num1 @ A3 @ B3 )
        = zero_z3563351764282998399l_num1 )
     => ( dvd_dv6812691276156420380l_num1 @ B3 @ A3 ) ) ).

% mod_0_imp_dvd
thf(fact_5699_mod__0__imp__dvd,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ B3 )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B3 @ A3 ) ) ).

% mod_0_imp_dvd
thf(fact_5700_mod__0__imp__dvd,axiom,
    ! [A3: int,B3: int] :
      ( ( ( modulo_modulo_int @ A3 @ B3 )
        = zero_zero_int )
     => ( dvd_dvd_int @ B3 @ A3 ) ) ).

% mod_0_imp_dvd
thf(fact_5701_mod__0__imp__dvd,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ B3 )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).

% mod_0_imp_dvd
thf(fact_5702_mod__0__imp__dvd,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ( modulo_modulo_uint32 @ A3 @ B3 )
        = zero_zero_uint32 )
     => ( dvd_dvd_uint32 @ B3 @ A3 ) ) ).

% mod_0_imp_dvd
thf(fact_5703_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A2 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5704_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A2: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A2 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5705_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A2 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5706_mod__eq__0__iff__dvd,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( modulo_modulo_nat @ A3 @ B3 )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B3 @ A3 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5707_mod__eq__0__iff__dvd,axiom,
    ! [A3: int,B3: int] :
      ( ( ( modulo_modulo_int @ A3 @ B3 )
        = zero_zero_int )
      = ( dvd_dvd_int @ B3 @ A3 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5708_mod__eq__0__iff__dvd,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ B3 )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B3 @ A3 ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5709_dvd__power__le,axiom,
    ! [X2: uint32,Y2: uint32,N2: nat,M: nat] :
      ( ( dvd_dvd_uint32 @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_uint32 @ ( power_power_uint32 @ X2 @ N2 ) @ ( power_power_uint32 @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5710_dvd__power__le,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1,N2: nat,M: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) @ ( power_2184487114949457152l_num1 @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5711_dvd__power__le,axiom,
    ! [X2: nat,Y2: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X2 @ N2 ) @ ( power_power_nat @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5712_dvd__power__le,axiom,
    ! [X2: int,Y2: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X2 @ N2 ) @ ( power_power_int @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5713_dvd__power__le,axiom,
    ! [X2: real,Y2: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X2 @ N2 ) @ ( power_power_real @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5714_dvd__power__le,axiom,
    ! [X2: complex,Y2: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X2 @ N2 ) @ ( power_power_complex @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5715_dvd__power__le,axiom,
    ! [X2: code_integer,Y2: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X2 @ Y2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ N2 ) @ ( power_8256067586552552935nteger @ Y2 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_5716_power__le__dvd,axiom,
    ! [A3: uint32,N2: nat,B3: uint32,M: nat] :
      ( ( dvd_dvd_uint32 @ ( power_power_uint32 @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5717_power__le__dvd,axiom,
    ! [A3: word_N3645301735248828278l_num1,N2: nat,B3: word_N3645301735248828278l_num1,M: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5718_power__le__dvd,axiom,
    ! [A3: nat,N2: nat,B3: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5719_power__le__dvd,axiom,
    ! [A3: int,N2: nat,B3: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5720_power__le__dvd,axiom,
    ! [A3: real,N2: nat,B3: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5721_power__le__dvd,axiom,
    ! [A3: complex,N2: nat,B3: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5722_power__le__dvd,axiom,
    ! [A3: code_integer,N2: nat,B3: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ B3 )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ M ) @ B3 ) ) ) ).

% power_le_dvd
thf(fact_5723_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: uint32] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_uint32 @ ( power_power_uint32 @ A3 @ M ) @ ( power_power_uint32 @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5724_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dv6812691276156420380l_num1 @ ( power_2184487114949457152l_num1 @ A3 @ M ) @ ( power_2184487114949457152l_num1 @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5725_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A3 @ M ) @ ( power_power_nat @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5726_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A3 @ M ) @ ( power_power_int @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5727_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A3 @ M ) @ ( power_power_real @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5728_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A3 @ M ) @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5729_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A3: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ M ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_5730_mod__eq__dvd__iff,axiom,
    ! [A3: int,C: int,B3: int] :
      ( ( ( modulo_modulo_int @ A3 @ C )
        = ( modulo_modulo_int @ B3 @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A3 @ B3 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_5731_mod__eq__dvd__iff,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ C )
        = ( modulo364778990260209775nteger @ B3 @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).

% mod_eq_dvd_iff
thf(fact_5732_dvd__minus__mod,axiom,
    ! [B3: nat,A3: nat] : ( dvd_dvd_nat @ B3 @ ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) ) ) ).

% dvd_minus_mod
thf(fact_5733_dvd__minus__mod,axiom,
    ! [B3: int,A3: int] : ( dvd_dvd_int @ B3 @ ( minus_minus_int @ A3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% dvd_minus_mod
thf(fact_5734_dvd__minus__mod,axiom,
    ! [B3: code_integer,A3: code_integer] : ( dvd_dvd_Code_integer @ B3 @ ( minus_8373710615458151222nteger @ A3 @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).

% dvd_minus_mod
thf(fact_5735_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_5736_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_5737_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_5738_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_5739_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_5740_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5741_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_5742_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_5743_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_5744_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_5745_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_5746_even__numeral,axiom,
    ! [N2: num] : ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_5747_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_5748_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_5749_unity__coeff__ex,axiom,
    ! [P: uint32 > $o,L2: uint32] :
      ( ( ? [X: uint32] : ( P @ ( times_times_uint32 @ L2 @ X ) ) )
      = ( ? [X: uint32] :
            ( ( dvd_dvd_uint32 @ L2 @ ( plus_plus_uint32 @ X @ zero_zero_uint32 ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5750_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L2: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5751_unity__coeff__ex,axiom,
    ! [P: word_N3645301735248828278l_num1 > $o,L2: word_N3645301735248828278l_num1] :
      ( ( ? [X: word_N3645301735248828278l_num1] : ( P @ ( times_7065122842183080059l_num1 @ L2 @ X ) ) )
      = ( ? [X: word_N3645301735248828278l_num1] :
            ( ( dvd_dv6812691276156420380l_num1 @ L2 @ ( plus_p361126936061061375l_num1 @ X @ zero_z3563351764282998399l_num1 ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5752_unity__coeff__ex,axiom,
    ! [P: real > $o,L2: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L2 @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5753_unity__coeff__ex,axiom,
    ! [P: rat > $o,L2: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L2 @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5754_unity__coeff__ex,axiom,
    ! [P: nat > $o,L2: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L2 @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5755_unity__coeff__ex,axiom,
    ! [P: int > $o,L2: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L2 @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_5756_unit__dvdE,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ~ ( ( A3 != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B3
             != ( times_3573771949741848930nteger @ A3 @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5757_unit__dvdE,axiom,
    ! [A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ~ ( ( A3 != zero_zero_nat )
         => ! [C2: nat] :
              ( B3
             != ( times_times_nat @ A3 @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5758_unit__dvdE,axiom,
    ! [A3: int,B3: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ~ ( ( A3 != zero_zero_int )
         => ! [C2: int] :
              ( B3
             != ( times_times_int @ A3 @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_5759_unit__div__eq__0__iff,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
          = zero_z3403309356797280102nteger )
        = ( A3 = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5760_unit__div__eq__0__iff,axiom,
    ! [B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A3 @ B3 )
          = zero_zero_nat )
        = ( A3 = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5761_unit__div__eq__0__iff,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( ( divide_divide_int @ A3 @ B3 )
          = zero_zero_int )
        = ( A3 = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5762_dvd__div__eq__mult,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A3 @ B3 )
       => ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
            = C )
          = ( B3
            = ( times_3573771949741848930nteger @ C @ A3 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5763_dvd__div__eq__mult,axiom,
    ! [A3: nat,B3: nat,C: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A3 @ B3 )
       => ( ( ( divide_divide_nat @ B3 @ A3 )
            = C )
          = ( B3
            = ( times_times_nat @ C @ A3 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5764_dvd__div__eq__mult,axiom,
    ! [A3: int,B3: int,C: int] :
      ( ( A3 != zero_zero_int )
     => ( ( dvd_dvd_int @ A3 @ B3 )
       => ( ( ( divide_divide_int @ B3 @ A3 )
            = C )
          = ( B3
            = ( times_times_int @ C @ A3 ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5765_div__dvd__iff__mult,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( B3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B3 @ A3 )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
          = ( dvd_dvd_Code_integer @ A3 @ ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5766_div__dvd__iff__mult,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B3 @ A3 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
          = ( dvd_dvd_nat @ A3 @ ( times_times_nat @ C @ B3 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5767_div__dvd__iff__mult,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( B3 != zero_zero_int )
     => ( ( dvd_dvd_int @ B3 @ A3 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
          = ( dvd_dvd_int @ A3 @ ( times_times_int @ C @ B3 ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5768_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B3 )
       => ( ( dvd_dvd_Code_integer @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5769_dvd__div__iff__mult,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B3 )
       => ( ( dvd_dvd_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A3 @ C ) @ B3 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5770_dvd__div__iff__mult,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B3 )
       => ( ( dvd_dvd_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A3 @ C ) @ B3 ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5771_dvd__div__div__eq__mult,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A3 @ B3 )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B3 @ A3 )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B3 @ C )
                = ( times_3573771949741848930nteger @ A3 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5772_dvd__div__div__eq__mult,axiom,
    ! [A3: nat,C: nat,B3: nat,D: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A3 @ B3 )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B3 @ A3 )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B3 @ C )
                = ( times_times_nat @ A3 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5773_dvd__div__div__eq__mult,axiom,
    ! [A3: int,C: int,B3: int,D: int] :
      ( ( A3 != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A3 @ B3 )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B3 @ A3 )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B3 @ C )
                = ( times_times_int @ A3 @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5774_unit__eq__div1,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A3 @ B3 )
          = C )
        = ( A3
          = ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ).

% unit_eq_div1
thf(fact_5775_unit__eq__div1,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( ( divide_divide_nat @ A3 @ B3 )
          = C )
        = ( A3
          = ( times_times_nat @ C @ B3 ) ) ) ) ).

% unit_eq_div1
thf(fact_5776_unit__eq__div1,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( ( divide_divide_int @ A3 @ B3 )
          = C )
        = ( A3
          = ( times_times_int @ C @ B3 ) ) ) ) ).

% unit_eq_div1
thf(fact_5777_unit__eq__div2,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( A3
          = ( divide6298287555418463151nteger @ C @ B3 ) )
        = ( ( times_3573771949741848930nteger @ A3 @ B3 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_5778_unit__eq__div2,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( A3
          = ( divide_divide_nat @ C @ B3 ) )
        = ( ( times_times_nat @ A3 @ B3 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_5779_unit__eq__div2,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( A3
          = ( divide_divide_int @ C @ B3 ) )
        = ( ( times_times_int @ A3 @ B3 )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_5780_div__mult__unit2,axiom,
    ! [C: code_integer,B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B3 @ A3 )
       => ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_5781_div__mult__unit2,axiom,
    ! [C: nat,B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B3 @ A3 )
       => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_5782_div__mult__unit2,axiom,
    ! [C: int,B3: int,A3: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B3 @ A3 )
       => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_5783_unit__div__commute,axiom,
    ! [B3: code_integer,A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ C ) @ B3 ) ) ) ).

% unit_div_commute
thf(fact_5784_unit__div__commute,axiom,
    ! [B3: nat,A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A3 @ C ) @ B3 ) ) ) ).

% unit_div_commute
thf(fact_5785_unit__div__commute,axiom,
    ! [B3: int,A3: int,C: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A3 @ B3 ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A3 @ C ) @ B3 ) ) ) ).

% unit_div_commute
thf(fact_5786_unit__div__mult__swap,axiom,
    ! [C: code_integer,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A3 @ ( divide6298287555418463151nteger @ B3 @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ B3 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_5787_unit__div__mult__swap,axiom,
    ! [C: nat,A3: nat,B3: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A3 @ ( divide_divide_nat @ B3 @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A3 @ B3 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_5788_unit__div__mult__swap,axiom,
    ! [C: int,A3: int,B3: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A3 @ ( divide_divide_int @ B3 @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A3 @ B3 ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_5789_is__unit__div__mult2__eq,axiom,
    ! [B3: code_integer,C: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5790_is__unit__div__mult2__eq,axiom,
    ! [B3: nat,C: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A3 @ B3 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5791_is__unit__div__mult2__eq,axiom,
    ! [B3: int,C: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A3 @ B3 ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5792_unit__imp__mod__eq__0,axiom,
    ! [B3: nat,A3: nat] :
      ( ( dvd_dvd_nat @ B3 @ one_one_nat )
     => ( ( modulo_modulo_nat @ A3 @ B3 )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_5793_unit__imp__mod__eq__0,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ one_one_int )
     => ( ( modulo_modulo_int @ A3 @ B3 )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_5794_unit__imp__mod__eq__0,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A3 @ B3 )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_5795_is__unit__power__iff,axiom,
    ! [A3: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N2 ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A3 @ one_one_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5796_is__unit__power__iff,axiom,
    ! [A3: int,N2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A3 @ N2 ) @ one_one_int )
      = ( ( dvd_dvd_int @ A3 @ one_one_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5797_is__unit__power__iff,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_5798_dvd__imp__le,axiom,
    ! [K: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% dvd_imp_le
thf(fact_5799_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_5800_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_5801_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_5802_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q2 )
          = ( modulo_modulo_nat @ N2 @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_5803_real__of__nat__div,axiom,
    ! [D: nat,N2: nat] :
      ( ( dvd_dvd_nat @ D @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_5804_even__zero,axiom,
    dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ zero_zero_uint32 ).

% even_zero
thf(fact_5805_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_5806_even__zero,axiom,
    dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ zero_z3563351764282998399l_num1 ).

% even_zero
thf(fact_5807_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_5808_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_5809_odd__even__add,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B3 )
       => ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( plus_plus_uint32 @ A3 @ B3 ) ) ) ) ).

% odd_even_add
thf(fact_5810_odd__even__add,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) ) ) ).

% odd_even_add
thf(fact_5811_odd__even__add,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B3 )
       => ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( plus_p361126936061061375l_num1 @ A3 @ B3 ) ) ) ) ).

% odd_even_add
thf(fact_5812_odd__even__add,axiom,
    ! [A3: nat,B3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A3 @ B3 ) ) ) ) ).

% odd_even_add
thf(fact_5813_odd__even__add,axiom,
    ! [A3: int,B3: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A3 @ B3 ) ) ) ) ).

% odd_even_add
thf(fact_5814_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_5815_odd__one,axiom,
    ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ one_one_uint32 ) ).

% odd_one
thf(fact_5816_odd__one,axiom,
    ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ one_on7727431528512463931l_num1 ) ).

% odd_one
thf(fact_5817_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_5818_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_5819_evenE,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: uint32] :
            ( A3
           != ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) ) ) ).

% evenE
thf(fact_5820_evenE,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: code_integer] :
            ( A3
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% evenE
thf(fact_5821_evenE,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: word_N3645301735248828278l_num1] :
            ( A3
           != ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B ) ) ) ).

% evenE
thf(fact_5822_evenE,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: nat] :
            ( A3
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% evenE
thf(fact_5823_evenE,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: int] :
            ( A3
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% evenE
thf(fact_5824_bit__eq__rec,axiom,
    ( ( ^ [Y5: code_integer,Z2: code_integer] : Y5 = Z2 )
    = ( ^ [A2: code_integer,B2: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5825_bit__eq__rec,axiom,
    ( ( ^ [Y5: word_N3645301735248828278l_num1,Z2: word_N3645301735248828278l_num1] : Y5 = Z2 )
    = ( ^ [A2: word_N3645301735248828278l_num1,B2: word_N3645301735248828278l_num1] :
          ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A2 )
            = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
            = ( divide1791077408188789448l_num1 @ B2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5826_bit__eq__rec,axiom,
    ( ( ^ [Y5: uint32,Z2: uint32] : Y5 = Z2 )
    = ( ^ [A2: uint32,B2: uint32] :
          ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A2 )
            = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
            = ( divide_divide_uint32 @ B2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5827_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5828_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [A2: int,B2: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5829_is__unitE,axiom,
    ! [A3: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A3 @ one_one_Code_integer )
     => ~ ( ( A3 != zero_z3403309356797280102nteger )
         => ! [B: code_integer] :
              ( ( B != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A3 )
                    = B )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B )
                      = A3 )
                   => ( ( ( times_3573771949741848930nteger @ A3 @ B )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A3 )
                       != ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5830_is__unitE,axiom,
    ! [A3: nat,C: nat] :
      ( ( dvd_dvd_nat @ A3 @ one_one_nat )
     => ~ ( ( A3 != zero_zero_nat )
         => ! [B: nat] :
              ( ( B != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A3 )
                    = B )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B )
                      = A3 )
                   => ( ( ( times_times_nat @ A3 @ B )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A3 )
                       != ( times_times_nat @ C @ B ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5831_is__unitE,axiom,
    ! [A3: int,C: int] :
      ( ( dvd_dvd_int @ A3 @ one_one_int )
     => ~ ( ( A3 != zero_zero_int )
         => ! [B: int] :
              ( ( B != zero_zero_int )
             => ( ( dvd_dvd_int @ B @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A3 )
                    = B )
                 => ( ( ( divide_divide_int @ one_one_int @ B )
                      = A3 )
                   => ( ( ( times_times_int @ A3 @ B )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A3 )
                       != ( times_times_int @ C @ B ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5832_is__unit__div__mult__cancel__left,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5833_is__unit__div__mult__cancel__left,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
       => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ A3 @ B3 ) )
          = ( divide_divide_nat @ one_one_nat @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5834_is__unit__div__mult__cancel__left,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( dvd_dvd_int @ B3 @ one_one_int )
       => ( ( divide_divide_int @ A3 @ ( times_times_int @ A3 @ B3 ) )
          = ( divide_divide_int @ one_one_int @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5835_is__unit__div__mult__cancel__right,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( A3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A3 @ ( times_3573771949741848930nteger @ B3 @ A3 ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5836_is__unit__div__mult__cancel__right,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
       => ( ( divide_divide_nat @ A3 @ ( times_times_nat @ B3 @ A3 ) )
          = ( divide_divide_nat @ one_one_nat @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5837_is__unit__div__mult__cancel__right,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( dvd_dvd_int @ B3 @ one_one_int )
       => ( ( divide_divide_int @ A3 @ ( times_times_int @ B3 @ A3 ) )
          = ( divide_divide_int @ one_one_int @ B3 ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5838_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( numera9087168376688890119uint32 @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5839_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5840_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( numera7442385471795722001l_num1 @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5841_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5842_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5843_dvd__power__iff,axiom,
    ! [X2: nat,M: nat,N2: nat] :
      ( ( X2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X2 @ M ) @ ( power_power_nat @ X2 @ N2 ) )
        = ( ( dvd_dvd_nat @ X2 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_5844_dvd__power__iff,axiom,
    ! [X2: int,M: nat,N2: nat] :
      ( ( X2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ N2 ) )
        = ( ( dvd_dvd_int @ X2 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_5845_dvd__power__iff,axiom,
    ! [X2: code_integer,M: nat,N2: nat] :
      ( ( X2 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X2 @ M ) @ ( power_8256067586552552935nteger @ X2 @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X2 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_5846_dvd__power,axiom,
    ! [N2: nat,X2: word_N3645301735248828278l_num1] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_on7727431528512463931l_num1 ) )
     => ( dvd_dv6812691276156420380l_num1 @ X2 @ ( power_2184487114949457152l_num1 @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5847_dvd__power,axiom,
    ! [N2: nat,X2: uint32] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_uint32 ) )
     => ( dvd_dvd_uint32 @ X2 @ ( power_power_uint32 @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5848_dvd__power,axiom,
    ! [N2: nat,X2: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_rat ) )
     => ( dvd_dvd_rat @ X2 @ ( power_power_rat @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5849_dvd__power,axiom,
    ! [N2: nat,X2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_nat ) )
     => ( dvd_dvd_nat @ X2 @ ( power_power_nat @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5850_dvd__power,axiom,
    ! [N2: nat,X2: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_int ) )
     => ( dvd_dvd_int @ X2 @ ( power_power_int @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5851_dvd__power,axiom,
    ! [N2: nat,X2: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_real ) )
     => ( dvd_dvd_real @ X2 @ ( power_power_real @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5852_dvd__power,axiom,
    ! [N2: nat,X2: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_complex ) )
     => ( dvd_dvd_complex @ X2 @ ( power_power_complex @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5853_dvd__power,axiom,
    ! [N2: nat,X2: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X2 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X2 @ ( power_8256067586552552935nteger @ X2 @ N2 ) ) ) ).

% dvd_power
thf(fact_5854_refines__If,axiom,
    ! [B3: $o,T2: heap_T2636463487746394924on_nat,T4: heap_T2636463487746394924on_nat,E2: heap_T2636463487746394924on_nat,E3: heap_T2636463487746394924on_nat] :
      ( ( B3
       => ( refine7594492741263601813on_nat @ T2 @ T4 ) )
     => ( ( ~ B3
         => ( refine7594492741263601813on_nat @ E2 @ E3 ) )
       => ( refine7594492741263601813on_nat @ ( if_Hea5867803462524415986on_nat @ B3 @ T2 @ E2 ) @ ( if_Hea5867803462524415986on_nat @ B3 @ T4 @ E3 ) ) ) ) ).

% refines_If
thf(fact_5855_refines__If,axiom,
    ! [B3: $o,T2: heap_Time_Heap_o,T4: heap_Time_Heap_o,E2: heap_Time_Heap_o,E3: heap_Time_Heap_o] :
      ( ( B3
       => ( refine_Imp_refines_o @ T2 @ T4 ) )
     => ( ( ~ B3
         => ( refine_Imp_refines_o @ E2 @ E3 ) )
       => ( refine_Imp_refines_o @ ( if_Heap_Time_Heap_o @ B3 @ T2 @ E2 ) @ ( if_Heap_Time_Heap_o @ B3 @ T4 @ E3 ) ) ) ) ).

% refines_If
thf(fact_5856_refines__If,axiom,
    ! [B3: $o,T2: heap_T8145700208782473153_VEBTi,T4: heap_T8145700208782473153_VEBTi,E2: heap_T8145700208782473153_VEBTi,E3: heap_T8145700208782473153_VEBTi] :
      ( ( B3
       => ( refine5565527176597971370_VEBTi @ T2 @ T4 ) )
     => ( ( ~ B3
         => ( refine5565527176597971370_VEBTi @ E2 @ E3 ) )
       => ( refine5565527176597971370_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ B3 @ T2 @ E2 ) @ ( if_Hea8453224502484754311_VEBTi @ B3 @ T4 @ E3 ) ) ) ) ).

% refines_If
thf(fact_5857_refines__refl,axiom,
    ! [P2: heap_T2636463487746394924on_nat] : ( refine7594492741263601813on_nat @ P2 @ P2 ) ).

% refines_refl
thf(fact_5858_refines__refl,axiom,
    ! [P2: heap_Time_Heap_o] : ( refine_Imp_refines_o @ P2 @ P2 ) ).

% refines_refl
thf(fact_5859_refines__refl,axiom,
    ! [P2: heap_T8145700208782473153_VEBTi] : ( refine5565527176597971370_VEBTi @ P2 @ P2 ) ).

% refines_refl
thf(fact_5860_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_5861_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_5862_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_5863_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_5864_dvd__minus__add,axiom,
    ! [Q2: nat,N2: nat,R: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N2 )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_5865_mod__nat__eqI,axiom,
    ! [R: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R @ N2 )
     => ( ( ord_less_eq_nat @ R @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R ) ) ) ) ).

% mod_nat_eqI
thf(fact_5866_diff__mod__le,axiom,
    ! [A3: nat,D: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ D )
     => ( ( dvd_dvd_nat @ B3 @ D )
       => ( ord_less_eq_nat @ ( minus_minus_nat @ A3 @ ( modulo_modulo_nat @ A3 @ B3 ) ) @ ( minus_minus_nat @ D @ B3 ) ) ) ) ).

% diff_mod_le
thf(fact_5867_even__two__times__div__two,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A3 ) ) ).

% even_two_times_div_two
thf(fact_5868_even__two__times__div__two,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A3 ) ) ).

% even_two_times_div_two
thf(fact_5869_even__two__times__div__two,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A3 ) ) ).

% even_two_times_div_two
thf(fact_5870_even__iff__mod__2__eq__zero,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
      = ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = zero_z3563351764282998399l_num1 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5871_even__iff__mod__2__eq__zero,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
      = ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_nat ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5872_even__iff__mod__2__eq__zero,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
      = ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5873_even__iff__mod__2__eq__zero,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
      = ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5874_even__iff__mod__2__eq__zero,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
      = ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = zero_zero_uint32 ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5875_odd__iff__mod__2__eq__one,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) )
      = ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
        = one_on7727431528512463931l_num1 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5876_odd__iff__mod__2__eq__one,axiom,
    ! [A3: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
      = ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5877_odd__iff__mod__2__eq__one,axiom,
    ! [A3: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
      = ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5878_odd__iff__mod__2__eq__one,axiom,
    ! [A3: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
      = ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5879_odd__iff__mod__2__eq__one,axiom,
    ! [A3: uint32] :
      ( ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) )
      = ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
        = one_one_uint32 ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_5880_power__mono__odd,axiom,
    ! [N2: nat,A3: real,B3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A3 @ B3 )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_5881_power__mono__odd,axiom,
    ! [N2: nat,A3: code_integer,B3: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_5882_power__mono__odd,axiom,
    ! [N2: nat,A3: rat,B3: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A3 @ B3 )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_5883_power__mono__odd,axiom,
    ! [N2: nat,A3: int,B3: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A3 @ B3 )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_5884_odd__pos,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% odd_pos
thf(fact_5885_even__set__bit__iff,axiom,
    ! [M: nat,A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se6647067497041451410uint32 @ M @ A3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5886_even__set__bit__iff,axiom,
    ! [M: nat,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5887_even__set__bit__iff,axiom,
    ! [M: nat,A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4894374433684937756l_num1 @ M @ A3 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5888_even__set__bit__iff,axiom,
    ! [M: nat,A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5889_even__set__bit__iff,axiom,
    ! [M: nat,A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5890_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_5891_oddE,axiom,
    ! [A3: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: code_integer] :
            ( A3
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_5892_oddE,axiom,
    ! [A3: uint32] :
      ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: uint32] :
            ( A3
           != ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ B ) @ one_one_uint32 ) ) ) ).

% oddE
thf(fact_5893_oddE,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: word_N3645301735248828278l_num1] :
            ( A3
           != ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ B ) @ one_on7727431528512463931l_num1 ) ) ) ).

% oddE
thf(fact_5894_oddE,axiom,
    ! [A3: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: nat] :
            ( A3
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_5895_oddE,axiom,
    ! [A3: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
     => ~ ! [B: int] :
            ( A3
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ one_one_int ) ) ) ).

% oddE
thf(fact_5896_mod2__eq__if,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = zero_z3563351764282998399l_num1 ) )
      & ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
          = one_on7727431528512463931l_num1 ) ) ) ).

% mod2_eq_if
thf(fact_5897_mod2__eq__if,axiom,
    ! [A3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = zero_zero_nat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
          = one_one_nat ) ) ) ).

% mod2_eq_if
thf(fact_5898_mod2__eq__if,axiom,
    ! [A3: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = zero_zero_int ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
          = one_one_int ) ) ) ).

% mod2_eq_if
thf(fact_5899_mod2__eq__if,axiom,
    ! [A3: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_5900_mod2__eq__if,axiom,
    ! [A3: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = zero_zero_uint32 ) )
      & ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
          = one_one_uint32 ) ) ) ).

% mod2_eq_if
thf(fact_5901_parity__cases,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
         != zero_z3563351764282998399l_num1 ) )
     => ~ ( ~ ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
         => ( ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
           != one_on7727431528512463931l_num1 ) ) ) ).

% parity_cases
thf(fact_5902_parity__cases,axiom,
    ! [A3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
         != zero_zero_nat ) )
     => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
         => ( ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
           != one_one_nat ) ) ) ).

% parity_cases
thf(fact_5903_parity__cases,axiom,
    ! [A3: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
         != zero_zero_int ) )
     => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
         => ( ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
           != one_one_int ) ) ) ).

% parity_cases
thf(fact_5904_parity__cases,axiom,
    ! [A3: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
         => ( ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_5905_parity__cases,axiom,
    ! [A3: uint32] :
      ( ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
       => ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
         != zero_zero_uint32 ) )
     => ~ ( ~ ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
         => ( ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
           != one_one_uint32 ) ) ) ).

% parity_cases
thf(fact_5906_zero__le__power__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ).

% zero_le_power_eq
thf(fact_5907_zero__le__power__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ) ) ).

% zero_le_power_eq
thf(fact_5908_zero__le__power__eq,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ).

% zero_le_power_eq
thf(fact_5909_zero__le__power__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ) ).

% zero_le_power_eq
thf(fact_5910_zero__le__odd__power,axiom,
    ! [N2: nat,A3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ).

% zero_le_odd_power
thf(fact_5911_zero__le__odd__power,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ) ).

% zero_le_odd_power
thf(fact_5912_zero__le__odd__power,axiom,
    ! [N2: nat,A3: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ).

% zero_le_odd_power
thf(fact_5913_zero__le__odd__power,axiom,
    ! [N2: nat,A3: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) )
        = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ) ).

% zero_le_odd_power
thf(fact_5914_zero__le__even__power,axiom,
    ! [N2: nat,A3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_5915_zero__le__even__power,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_5916_zero__le__even__power,axiom,
    ! [N2: nat,A3: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_5917_zero__le__even__power,axiom,
    ! [N2: nat,A3: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_5918_divides__aux__def,axiom,
    ( unique6322359934112328802ux_nat
    = ( ^ [Qr: product_prod_nat_nat] :
          ( ( product_snd_nat_nat @ Qr )
          = zero_zero_nat ) ) ) ).

% divides_aux_def
thf(fact_5919_divides__aux__def,axiom,
    ( unique6319869463603278526ux_int
    = ( ^ [Qr: product_prod_int_int] :
          ( ( product_snd_int_int @ Qr )
          = zero_zero_int ) ) ) ).

% divides_aux_def
thf(fact_5920_zero__less__power__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A3 != zero_z3403309356797280102nteger ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 ) ) ) ) ).

% zero_less_power_eq
thf(fact_5921_zero__less__power__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A3 @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A3 != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ).

% zero_less_power_eq
thf(fact_5922_zero__less__power__eq,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A3 @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A3 != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ).

% zero_less_power_eq
thf(fact_5923_zero__less__power__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A3 @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A3 != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_int @ zero_zero_int @ A3 ) ) ) ) ).

% zero_less_power_eq
thf(fact_5924_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_5925_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_5926_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_5927_power__le__zero__eq,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_real @ A3 @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A3 = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5928_power__le__zero__eq,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ zero_z3403309356797280102nteger )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A3 = zero_z3403309356797280102nteger ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5929_power__le__zero__eq,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A3 @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A3 = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5930_power__le__zero__eq,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_int @ A3 @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A3 = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_5931_even__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( suc @ zero_zero_nat ) )
     => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% even_mod_4_div_2
thf(fact_5932_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_5933_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ ( minus_4019991460397169231l_num1 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) @ one_on7727431528512463931l_num1 ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
          = zero_z3563351764282998399l_num1 )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_5934_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( minus_minus_uint32 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) @ one_one_uint32 ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_uint32 )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_5935_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_5936_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_5937_odd__mod__4__div__2,axiom,
    ! [N2: nat] :
      ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
     => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% odd_mod_4_div_2
thf(fact_5938_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: nat > heap_Time_Heap_o,F3: nat > heap_Time_Heap_o] :
      ( ( refine1365783493865988805es_nat @ M @ M6 )
     => ( ! [X3: nat] : ( refine_Imp_refines_o @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine_Imp_refines_o @ ( heap_Time_bind_nat_o @ M @ F ) @ ( heap_Time_bind_nat_o @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5939_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: nat > heap_T8145700208782473153_VEBTi,F3: nat > heap_T8145700208782473153_VEBTi] :
      ( ( refine1365783493865988805es_nat @ M @ M6 )
     => ( ! [X3: nat] : ( refine5565527176597971370_VEBTi @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine5565527176597971370_VEBTi @ ( heap_T844888390831797134_VEBTi @ M @ F ) @ ( heap_T844888390831797134_VEBTi @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5940_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: $o > heap_Time_Heap_o,F3: $o > heap_Time_Heap_o] :
      ( ( refine_Imp_refines_o @ M @ M6 )
     => ( ! [X3: $o] : ( refine_Imp_refines_o @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine_Imp_refines_o @ ( heap_Time_bind_o_o @ M @ F ) @ ( heap_Time_bind_o_o @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5941_refines__bind,axiom,
    ! [M: heap_Time_Heap_o,M6: heap_Time_Heap_o,F: $o > heap_T8145700208782473153_VEBTi,F3: $o > heap_T8145700208782473153_VEBTi] :
      ( ( refine_Imp_refines_o @ M @ M6 )
     => ( ! [X3: $o] : ( refine5565527176597971370_VEBTi @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine5565527176597971370_VEBTi @ ( heap_T5998771940306268294_VEBTi @ M @ F ) @ ( heap_T5998771940306268294_VEBTi @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5942_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M6: heap_T8145700208782473153_VEBTi,F: vEBT_VEBTi > heap_Time_Heap_o,F3: vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( refine5565527176597971370_VEBTi @ M @ M6 )
     => ( ! [X3: vEBT_VEBTi] : ( refine_Imp_refines_o @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine_Imp_refines_o @ ( heap_T3040810144269856602EBTi_o @ M @ F ) @ ( heap_T3040810144269856602EBTi_o @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5943_refines__bind,axiom,
    ! [M: heap_T8145700208782473153_VEBTi,M6: heap_T8145700208782473153_VEBTi,F: vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F3: vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( refine5565527176597971370_VEBTi @ M @ M6 )
     => ( ! [X3: vEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine5565527176597971370_VEBTi @ ( heap_T1006145433769338483_VEBTi @ M @ F ) @ ( heap_T1006145433769338483_VEBTi @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5944_refines__bind,axiom,
    ! [M: heap_Time_Heap_nat,M6: heap_Time_Heap_nat,F: nat > heap_T2636463487746394924on_nat,F3: nat > heap_T2636463487746394924on_nat] :
      ( ( refine1365783493865988805es_nat @ M @ M6 )
     => ( ! [X3: nat] : ( refine7594492741263601813on_nat @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine7594492741263601813on_nat @ ( heap_T8222160169144143993on_nat @ M @ F ) @ ( heap_T8222160169144143993on_nat @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5945_refines__bind,axiom,
    ! [M: heap_T4980287057938770641_VEBTi,M6: heap_T4980287057938770641_VEBTi,F: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi,F3: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( refine3700189196150522554_VEBTi @ M @ M6 )
     => ( ! [X3: list_VEBT_VEBTi] : ( refine5565527176597971370_VEBTi @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine5565527176597971370_VEBTi @ ( heap_T5877712393672139267_VEBTi @ M @ F ) @ ( heap_T5877712393672139267_VEBTi @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5946_refines__bind,axiom,
    ! [M: heap_T2636463487746394924on_nat,M6: heap_T2636463487746394924on_nat,F: option_nat > heap_Time_Heap_o,F3: option_nat > heap_Time_Heap_o] :
      ( ( refine7594492741263601813on_nat @ M @ M6 )
     => ( ! [X3: option_nat] : ( refine_Imp_refines_o @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine_Imp_refines_o @ ( heap_T6471384023045698863_nat_o @ M @ F ) @ ( heap_T6471384023045698863_nat_o @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5947_refines__bind,axiom,
    ! [M: heap_T2636463487746394924on_nat,M6: heap_T2636463487746394924on_nat,F: option_nat > heap_T8145700208782473153_VEBTi,F3: option_nat > heap_T8145700208782473153_VEBTi] :
      ( ( refine7594492741263601813on_nat @ M @ M6 )
     => ( ! [X3: option_nat] : ( refine5565527176597971370_VEBTi @ ( F @ X3 ) @ ( F3 @ X3 ) )
       => ( refine5565527176597971370_VEBTi @ ( heap_T5661892481228163294_VEBTi @ M @ F ) @ ( heap_T5661892481228163294_VEBTi @ M6 @ F3 ) ) ) ) ).

% refines_bind
thf(fact_5948_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_5949_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
thf(fact_5950_even__mult__exp__div__exp__iff,axiom,
    ! [A3: code_integer,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5951_even__mult__exp__div__exp__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,M: nat,N2: nat] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ ( times_7065122842183080059l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ M ) ) @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ N2 )
          = zero_z3563351764282998399l_num1 )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A3 @ ( power_2184487114949457152l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5952_even__mult__exp__div__exp__iff,axiom,
    ! [A3: uint32,M: nat,N2: nat] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ ( times_times_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_uint32 )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A3 @ ( power_power_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5953_even__mult__exp__div__exp__iff,axiom,
    ! [A3: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5954_even__mult__exp__div__exp__iff,axiom,
    ! [A3: int,M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5955_refines__option,axiom,
    ! [A3: option4927543243414619207at_nat,A5: option4927543243414619207at_nat,M1: heap_Time_Heap_o,M12: heap_Time_Heap_o,M22: product_prod_nat_nat > heap_Time_Heap_o,M23: product_prod_nat_nat > heap_Time_Heap_o] :
      ( ( A3 = A5 )
     => ( ( refine_Imp_refines_o @ M1 @ M12 )
       => ( ! [X3: product_prod_nat_nat] : ( refine_Imp_refines_o @ ( M22 @ X3 ) @ ( M23 @ X3 ) )
         => ( refine_Imp_refines_o @ ( case_o1442776274061689234at_nat @ M1 @ M22 @ A3 ) @ ( case_o1442776274061689234at_nat @ M12 @ M23 @ A5 ) ) ) ) ) ).

% refines_option
thf(fact_5956_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( suc @ zero_zero_nat ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( suc @ zero_zero_nat ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
thf(fact_5957_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.simps(3)
thf(fact_5958_VEBT__internal_OTb_H_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
thf(fact_5959_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_VEBT_Tb @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.simps(3)
thf(fact_5960_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
thf(fact_5961_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
thf(fact_5962_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( suc @ Va2 ) ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
thf(fact_5963_VEBT__internal_OTb_Oelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_VEBT_Tb @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( numeral_numeral_int @ ( bit1 @ one ) ) ) )
         => ~ ! [N4: nat] :
                ( ( X2
                  = ( suc @ ( suc @ N4 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                     => ( Y2
                        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
thf(fact_5964_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( numeral_numeral_nat @ ( bit1 @ one ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
thf(fact_5965_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
thf(fact_5966_pow__divides__pow__iff,axiom,
    ! [N2: nat,A3: nat,B3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A3 @ N2 ) @ ( power_power_nat @ B3 @ N2 ) )
        = ( dvd_dvd_nat @ A3 @ B3 ) ) ) ).

% pow_divides_pow_iff
thf(fact_5967_pow__divides__pow__iff,axiom,
    ! [N2: nat,A3: int,B3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) )
        = ( dvd_dvd_int @ A3 @ B3 ) ) ) ).

% pow_divides_pow_iff
thf(fact_5968_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q4: code_integer,R6: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R6 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R6 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R6 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_5969_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5970_div2__even__ext__nat,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y2 ) )
       => ( X2 = Y2 ) ) ) ).

% div2_even_ext_nat
thf(fact_5971_bezout__add__strong__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D2 @ A3 )
          & ( dvd_dvd_nat @ D2 @ B3 )
          & ( ( times_times_nat @ A3 @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y3 ) @ D2 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_5972_flip__bit__Suc,axiom,
    ! [N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( bit_se4491814353640558621l_num1 @ ( suc @ N2 ) @ A3 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5973_flip__bit__Suc,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A3 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5974_flip__bit__Suc,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( bit_se7025624438249859091uint32 @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ N2 @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5975_flip__bit__Suc,axiom,
    ! [N2: nat,A3: int] :
      ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5976_flip__bit__Suc,axiom,
    ! [N2: nat,A3: nat] :
      ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_5977_ln__inj__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ( ln_ln_real @ X2 )
            = ( ln_ln_real @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% ln_inj_iff
thf(fact_5978_ln__less__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5979_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_5980_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_5981_ln__le__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_5982_ln__eq__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = zero_zero_real )
        = ( X2 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_5983_ln__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_iff
thf(fact_5984_ln__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_5985_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5986_ln__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_5987_ln__ge__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_iff
thf(fact_5988_ln__less__self,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_less_self
thf(fact_5989_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_5990_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_5991_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_5992_zdvd__mono,axiom,
    ! [K: int,M: int,T2: int] :
      ( ( K != zero_zero_int )
     => ( ( dvd_dvd_int @ M @ T2 )
        = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T2 ) ) ) ) ).

% zdvd_mono
thf(fact_5993_ln__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).

% ln_bound
thf(fact_5994_ln__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_gt_zero
thf(fact_5995_ln__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_5996_ln__gt__zero__imp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_5997_ln__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).

% ln_ge_zero
thf(fact_5998_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( ( dvd_dvd_int @ Z @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_5999_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_6000_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_6001_real__of__int__div,axiom,
    ! [D: int,N2: int] :
      ( ( dvd_dvd_int @ D @ N2 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_6002_zero__integer_Orsp,axiom,
    zero_zero_int = zero_zero_int ).

% zero_integer.rsp
thf(fact_6003_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
thf(fact_6004_prod__decode__aux_Ocases,axiom,
    ! [X2: product_prod_nat_nat] :
      ~ ! [K2: nat,M4: nat] :
          ( X2
         != ( product_Pair_nat_nat @ K2 @ M4 ) ) ).

% prod_decode_aux.cases
thf(fact_6005_times__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_6006_times__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_6007_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_6008_ln__ge__zero__imp__ge__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_6009_ln__add__one__self__le__self,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self
thf(fact_6010_ln__mult,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y2 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).

% ln_mult
thf(fact_6011_ln__eq__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ln_ln_real @ X2 )
          = ( minus_minus_real @ X2 @ one_one_real ) )
       => ( X2 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_6012_ln__div,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y2 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) ) ) ) ).

% ln_div
thf(fact_6013_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_6014_int__div__sub__1,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ one_one_int @ M )
     => ( ( ( dvd_dvd_int @ M @ N2 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N2 @ one_one_int ) @ M )
            = ( minus_minus_int @ ( divide_divide_int @ N2 @ M ) @ one_one_int ) ) )
        & ( ~ ( dvd_dvd_int @ M @ N2 )
         => ( ( divide_divide_int @ ( minus_minus_int @ N2 @ one_one_int ) @ M )
            = ( divide_divide_int @ N2 @ M ) ) ) ) ) ).

% int_div_sub_1
thf(fact_6015_bset_I9_J,axiom,
    ! [D: int,D4: int,B6: set_int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B6 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% bset(9)
thf(fact_6016_bset_I10_J,axiom,
    ! [D: int,D4: int,B6: set_int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B6 )
                 => ( X4
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% bset(10)
thf(fact_6017_aset_I9_J,axiom,
    ! [D: int,D4: int,A4: set_int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
           => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(9)
thf(fact_6018_aset_I10_J,axiom,
    ! [D: int,D4: int,A4: set_int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A4 )
                 => ( X4
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X4 @ T2 ) )
           => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(10)
thf(fact_6019_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_6020_ln__le__minus__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_6021_ln__diff__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y2 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y2 ) @ Y2 ) ) ) ) ).

% ln_diff_le
thf(fact_6022_ln__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( power_power_real @ X2 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_realpow
thf(fact_6023_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_6024_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B3 )
         => ( ( B3 != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( ( log @ A3 @ X2 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( ln_ln_real @ A3 ) ) @ ( log @ B3 @ X2 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_6025_list__decode_Ocases,axiom,
    ! [X2: nat] :
      ( ( X2 != zero_zero_nat )
     => ~ ! [N4: nat] :
            ( X2
           != ( suc @ N4 ) ) ) ).

% list_decode.cases
thf(fact_6026_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A3: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( P @ A @ B )
          = ( P @ B @ A ) )
     => ( ! [A: nat] : ( P @ A @ zero_zero_nat )
       => ( ! [A: nat,B: nat] :
              ( ( P @ A @ B )
             => ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
         => ( P @ A3 @ B3 ) ) ) ) ).

% Euclid_induct
thf(fact_6027_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : ( dvd_dvd_nat @ A3 @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_6028_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
        & ( zero_zero_nat != A3 ) ) ).

% gcd_nat.extremum_strict
thf(fact_6029_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
      = ( A3 = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_6030_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A3 @ zero_zero_nat )
        & ( A3 != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_6031_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A3 )
     => ( A3 = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_6032_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_6033_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% plus_integer_code(1)
thf(fact_6034_plus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
      = L2 ) ).

% plus_integer_code(2)
thf(fact_6035_eme1p,axiom,
    ! [N2: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ N2 ) @ D )
            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ N2 @ D ) ) ) ) ) ) ).

% eme1p
thf(fact_6036_emep1,axiom,
    ! [N2: int,D: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ D )
         => ( ( modulo_modulo_int @ ( plus_plus_int @ N2 @ one_one_int ) @ D )
            = ( plus_plus_int @ ( modulo_modulo_int @ N2 @ D ) @ one_one_int ) ) ) ) ) ).

% emep1
thf(fact_6037_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] :
      ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
      = K ) ).

% minus_integer_code(1)
thf(fact_6038_even__flip__bit__iff,axiom,
    ! [M: nat,A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se7025624438249859091uint32 @ M @ A3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6039_even__flip__bit__iff,axiom,
    ! [M: nat,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6040_even__flip__bit__iff,axiom,
    ! [M: nat,A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se4491814353640558621l_num1 @ M @ A3 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6041_even__flip__bit__iff,axiom,
    ! [M: nat,A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6042_even__flip__bit__iff,axiom,
    ! [M: nat,A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_6043_ln__one__plus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6044_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_6045_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
     => ( ! [M4: nat,N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
             => ( P @ M4 @ N4 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_6046_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ( vEBT_V854960066525838166emberi
    = ( ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6104975476656191286Heap_o
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
              @ ^ [Uu: product_unit] :
                  ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                  @ ( produc3505292621261808240Heap_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                        @ ( if_Heap_Time_Heap_o @ ( X = Mi3 ) @ ( heap_Time_return_o @ $true )
                          @ ( if_Heap_Time_Heap_o @ ( X = Ma3 ) @ ( heap_Time_return_o @ $true )
                            @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_o @ $false )
                              @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X ) @ ( heap_Time_return_o @ $false )
                                @ ( produc1330493526443650053Heap_o
                                  @ ^ [Info3: option4927543243414619207at_nat] :
                                      ( produc5946672270950774087Heap_o
                                      @ ^ [Deg3: nat] :
                                          ( produc5048428016959714504Heap_o
                                          @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                              ( heap_T7577940988442287570unit_o
                                              @ ( refine_Imp_assert
                                                @ ( ( Info2 = Info3 )
                                                  & ( Deg2 = Deg3 ) ) )
                                              @ ^ [Uv: product_unit] :
                                                  ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [L: nat] :
                                                          ( heap_T7577940988442287570unit_o
                                                          @ ( refine_Imp_assert
                                                            @ ( ( L
                                                                = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                              & ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                          @ ^ [Uw: product_unit] :
                                                              ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                              @ ^ [Len: nat] :
                                                                  ( heap_T7577940988442287570unit_o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( Len
                                                                      = ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                  @ ^ [Ux: product_unit] :
                                                                      ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                                      @ ( heap_T7577940988442287570unit_o
                                                                        @ ( refine_Imp_assert
                                                                          @ ( ( H
                                                                              = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                            & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                                                                        @ ^ [Uy: product_unit] :
                                                                            ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_V854960066525838166emberi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Th @ L ) ) )
                                                                      @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                  @ ( vEBT_c634343235235684882T_VEBT
                                    @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                    @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                                    @ T ) ) ) ) ) ) ) )
                  @ Info2 ) )
          @ ^ [A2: $o,B2: $o] :
              ( heap_Time_return_o
              @ ( ( ( X = zero_zero_nat )
                 => A2 )
                & ( ( X != zero_zero_nat )
                 => ( ( ( X = one_one_nat )
                     => B2 )
                    & ( X = one_one_nat ) ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_memberi'.simps
thf(fact_6047_vebt__memberi_Osimps,axiom,
    ( vEBT_vebt_memberi
    = ( ^ [T: vEBT_VEBTi,X: nat] :
          ( vEBT_c6104975476656191286Heap_o
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
              @ ( produc3505292621261808240Heap_o
                @ ^ [Mi3: nat,Ma3: nat] :
                    ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                    @ ( if_Heap_Time_Heap_o @ ( X = Mi3 ) @ ( heap_Time_return_o @ $true )
                      @ ( if_Heap_Time_Heap_o @ ( X = Ma3 ) @ ( heap_Time_return_o @ $true )
                        @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_o @ $false )
                          @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X ) @ ( heap_Time_return_o @ $false )
                            @ ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [L: nat] :
                                      ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList )
                                      @ ^ [Len: nat] :
                                          ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                          @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList @ H )
                                            @ ^ [Th: vEBT_VEBTi] : ( vEBT_vebt_memberi @ Th @ L ) )
                                          @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
              @ Info2 )
          @ ^ [A2: $o,B2: $o] :
              ( heap_Time_return_o
              @ ( ( ( X = zero_zero_nat )
                 => A2 )
                & ( ( X != zero_zero_nat )
                 => ( ( ( X = one_one_nat )
                     => B2 )
                    & ( X = one_one_nat ) ) ) ) )
          @ T ) ) ) ).

% vebt_memberi.simps
thf(fact_6048_VEBT__internal_Ovebt__memberi_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o > nat > $o,Xa: produc3960855945107176009Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat] :
      ( ! [Vebt_memberi: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: $o,N5: nat] :
              ( ( heap_Time_effect_o @ ( Vebt_memberi @ A6 @ B5 @ Ba ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ Ba @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: $o,N4: nat] :
              ( ( heap_Time_effect_o
                @ ( vEBT_c6104975476656191286Heap_o
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                      @ ^ [Uu: product_unit] :
                          ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                          @ ( produc3505292621261808240Heap_o
                            @ ^ [Mi3: nat,Ma3: nat] :
                                ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                                @ ( if_Heap_Time_Heap_o @ ( X3 = Mi3 ) @ ( heap_Time_return_o @ $true )
                                  @ ( if_Heap_Time_Heap_o @ ( X3 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                    @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                      @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X3 ) @ ( heap_Time_return_o @ $false )
                                        @ ( produc1330493526443650053Heap_o
                                          @ ^ [Info3: option4927543243414619207at_nat] :
                                              ( produc5946672270950774087Heap_o
                                              @ ^ [Deg3: nat] :
                                                  ( produc5048428016959714504Heap_o
                                                  @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                                      ( heap_T7577940988442287570unit_o
                                                      @ ( refine_Imp_assert
                                                        @ ( ( Info2 = Info3 )
                                                          & ( Deg2 = Deg3 ) ) )
                                                      @ ^ [Uv: product_unit] :
                                                          ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                              @ ^ [L: nat] :
                                                                  ( heap_T7577940988442287570unit_o
                                                                  @ ( refine_Imp_assert
                                                                    @ ( ( L
                                                                        = ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                      & ( H
                                                                        = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                  @ ^ [Uw: product_unit] :
                                                                      ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                      @ ^ [Len: nat] :
                                                                          ( heap_T7577940988442287570unit_o
                                                                          @ ( refine_Imp_assert
                                                                            @ ( Len
                                                                              = ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                          @ ^ [Ux: product_unit] :
                                                                              ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                                              @ ( heap_T7577940988442287570unit_o
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( ( H
                                                                                      = ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                    & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                                                                                @ ^ [Uy: product_unit] :
                                                                                    ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Th @ L ) ) )
                                                                              @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                          @ ( vEBT_c634343235235684882T_VEBT
                                            @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                            @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                                            @ T3 ) ) ) ) ) ) ) )
                          @ Info2 ) )
                  @ ^ [A2: $o,B2: $o] :
                      ( heap_Time_return_o
                      @ ( ( ( X3 = zero_zero_nat )
                         => A2 )
                        & ( ( X3 != zero_zero_nat )
                         => ( ( ( X3 = one_one_nat )
                             => B2 )
                            & ( X3 = one_one_nat ) ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa2
                @ N4 )
             => ( P @ T3 @ Ti4 @ X3 @ Ta @ Tia @ Xa2 @ N4 ) ) )
     => ( ( heap_Time_effect_o @ ( produc5872130906356439992Heap_o @ ( produc2327743382103342416Heap_o @ vEBT_V854960066525838166emberi ) @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc1340562934675340024_nat_o @ ( produc3077134696498096400_nat_o @ P ) @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
thf(fact_6049_bezw_Oelims,axiom,
    ! [X2: nat,Xa: nat,Y2: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa )
        = Y2 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y2
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y2
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_6050_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_6051_unset__bit__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ zero_zero_nat @ A3 )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6052_unset__bit__0,axiom,
    ! [A3: uint32] :
      ( ( bit_se4315839071623982667uint32 @ zero_zero_nat @ A3 )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6053_unset__bit__0,axiom,
    ! [A3: int] :
      ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A3 )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6054_unset__bit__0,axiom,
    ! [A3: nat] :
      ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A3 )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_6055_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_6056_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_6057_bezw__0,axiom,
    ! [X2: nat] :
      ( ( bezw @ X2 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_6058_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_6059_time__array__len,axiom,
    ! [P2: array_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_nat @ ( array_len_VEBT_VEBTi @ P2 ) @ H2 )
      = one_one_nat ) ).

% time_array_len
thf(fact_6060_TBOUND__len,axiom,
    ! [Xs2: array_VEBT_VEBTi] : ( time_TBOUND_nat @ ( array_len_VEBT_VEBTi @ Xs2 ) @ one_one_nat ) ).

% TBOUND_len
thf(fact_6061_vebt__memberi_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o > nat > $o,Xa: produc3881548065746020326Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat] :
      ( ! [Vebt_memberi2: vEBT_VEBTi > nat > heap_Time_Heap_o] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: $o,N5: nat] :
              ( ( heap_Time_effect_o @ ( Vebt_memberi2 @ A6 @ B5 ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: $o,N4: nat] :
              ( ( heap_Time_effect_o
                @ ( vEBT_c6104975476656191286Heap_o
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                      @ ( produc3505292621261808240Heap_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                            @ ( if_Heap_Time_Heap_o @ ( X3 = Mi3 ) @ ( heap_Time_return_o @ $true )
                              @ ( if_Heap_Time_Heap_o @ ( X3 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                  @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X3 ) @ ( heap_Time_return_o @ $false )
                                    @ ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                          @ ^ [L: nat] :
                                              ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList )
                                              @ ^ [Len: nat] :
                                                  ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                  @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList @ H )
                                                    @ ^ [Th: vEBT_VEBTi] : ( Vebt_memberi2 @ Th @ L ) )
                                                  @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                      @ Info2 )
                  @ ^ [A2: $o,B2: $o] :
                      ( heap_Time_return_o
                      @ ( ( ( X3 = zero_zero_nat )
                         => A2 )
                        & ( ( X3 != zero_zero_nat )
                         => ( ( ( X3 = one_one_nat )
                             => B2 )
                            & ( X3 = one_one_nat ) ) ) ) )
                  @ T3 )
                @ Ta
                @ Xa2
                @ R3
                @ N4 )
             => ( P @ T3 @ X3 @ Ta @ Xa2 @ R3 @ N4 ) ) )
     => ( ( heap_Time_effect_o @ ( produc770043135277712853Heap_o @ vEBT_vebt_memberi @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc1840203461219862933_nat_o @ P @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% vebt_memberi.raw_induct
thf(fact_6062_even__unset__bit__iff,axiom,
    ! [M: nat,A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ M @ A3 ) )
      = ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6063_even__unset__bit__iff,axiom,
    ! [M: nat,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A3 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6064_even__unset__bit__iff,axiom,
    ! [M: nat,A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ M @ A3 ) )
      = ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6065_even__unset__bit__iff,axiom,
    ! [M: nat,A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A3 ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6066_even__unset__bit__iff,axiom,
    ! [M: nat,A3: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A3 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_6067_bezw__non__0,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y2 )
     => ( ( bezw @ X2 @ Y2 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_6068_unset__bit__Suc,axiom,
    ! [N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( bit_se5331074070815623765l_num1 @ ( suc @ N2 ) @ A3 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_se5331074070815623765l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6069_unset__bit__Suc,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A3 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6070_unset__bit__Suc,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( bit_se4315839071623982667uint32 @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_se4315839071623982667uint32 @ N2 @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6071_unset__bit__Suc,axiom,
    ! [N2: nat,A3: int] :
      ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6072_unset__bit__Suc,axiom,
    ! [N2: nat,A3: nat] :
      ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_nat @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_6073_tanh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6074_VEBT__internal_Ovebt__succi_H_Omono,axiom,
    ! [X2: produc3960855945107176009Ti_nat] :
      ( comple6977564771798581627on_nat @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
      @ ^ [Vebt_succi3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
          ( produc183673358652719894on_nat
          @ ( produc1061038227461121684on_nat
            @ ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6250501799366334488on_nat
                @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
                    @ ^ [Uu: product_unit] :
                        ( produc2190226783763740553on_nat
                        @ ^ [Info3: option4927543243414619207at_nat] :
                            ( produc2606485630176857543on_nat
                            @ ^ [Deg3: nat] :
                                ( produc6124225815318652422on_nat
                                @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                    ( heap_T5999496708990702694on_nat
                                    @ ( refine_Imp_assert
                                      @ ( ( Info3 = Info2 )
                                        & ( Deg3 = Deg2 )
                                        & ( vEBT_is_Node @ T ) ) )
                                    @ ^ [Uv: product_unit] :
                                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                        @ ^ [Mima: product_prod_nat_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [L: nat] :
                                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [H: nat] :
                                                          ( heap_T5999496708990702694on_nat
                                                          @ ( refine_Imp_assert
                                                            @ ( L
                                                              = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                          @ ^ [Uw: product_unit] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( H
                                                                  = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Ux: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                  @ ^ [Uy: product_unit] :
                                                                      ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                      @ ^ [Aktnode: vEBT_VEBTi] :
                                                                          ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                                                          @ ^ [Maxlow: option_nat] :
                                                                              ( heap_T5999496708990702694on_nat
                                                                              @ ( refine_Imp_assert
                                                                                @ ( Maxlow
                                                                                  = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                              @ ^ [Uz: product_unit] :
                                                                                  ( if_Hea5867803462524415986on_nat
                                                                                  @ ( ( Maxlow != none_nat )
                                                                                    & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_succi3 ) @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                    @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                  @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_succi3 ) @ Summary3 @ Summary2 @ H )
                                                                                    @ ^ [Succsum: option_nat] :
                                                                                        ( heap_T5999496708990702694on_nat
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( ( Succsum = none_nat )
                                                                                            = ( ( vEBT_vebt_succ @ Summary3 @ H )
                                                                                              = none_nat ) ) )
                                                                                        @ ^ [Va: product_unit] :
                                                                                            ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                            @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                                                              @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                  ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                  @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info2 ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                          @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                          @ T ) ) )
                @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_succi'.mono
thf(fact_6075_VEBT__internal_Ovebt__predi_H_Omono,axiom,
    ! [X2: produc3960855945107176009Ti_nat] :
      ( comple6977564771798581627on_nat @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
      @ ^ [Vebt_predi3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
          ( produc183673358652719894on_nat
          @ ( produc1061038227461121684on_nat
            @ ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6250501799366334488on_nat
                @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
                    @ ^ [Uu: product_unit] :
                        ( produc2190226783763740553on_nat
                        @ ^ [Info3: option4927543243414619207at_nat] :
                            ( produc2606485630176857543on_nat
                            @ ^ [Deg3: nat] :
                                ( produc6124225815318652422on_nat
                                @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                    ( heap_T5999496708990702694on_nat
                                    @ ( refine_Imp_assert
                                      @ ( ( Info3 = Info2 )
                                        & ( Deg3 = Deg2 )
                                        & ( vEBT_is_Node @ T ) ) )
                                    @ ^ [Uv: product_unit] :
                                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                        @ ^ [Mima: product_prod_nat_nat] :
                                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [L: nat] :
                                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [H: nat] :
                                                        ( heap_T5999496708990702694on_nat
                                                        @ ( refine_Imp_assert
                                                          @ ( L
                                                            = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                        @ ^ [Uw: product_unit] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( H
                                                                = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Ux: product_unit] :
                                                                ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                @ ^ [Uy: product_unit] :
                                                                    ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                    @ ^ [Aktnode: vEBT_VEBTi] :
                                                                        ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                                                        @ ^ [Minlow: option_nat] :
                                                                            ( heap_T5999496708990702694on_nat
                                                                            @ ( refine_Imp_assert
                                                                              @ ( Minlow
                                                                                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                            @ ^ [Uz: product_unit] :
                                                                                ( if_Hea5867803462524415986on_nat
                                                                                @ ( ( Minlow != none_nat )
                                                                                  & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                                                @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                  @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                @ ( heap_T3669509953089699273on_nat @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) @ Summary3 @ Summary2 @ H )
                                                                                  @ ^ [Predsum: option_nat] :
                                                                                      ( heap_T5999496708990702694on_nat
                                                                                      @ ( refine_Imp_assert
                                                                                        @ ( ( Predsum = none_nat )
                                                                                          = ( ( vEBT_vebt_pred @ Summary3 @ H )
                                                                                            = none_nat ) ) )
                                                                                      @ ^ [Va: product_unit] :
                                                                                          ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                          @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                                                            @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                        @ Info2 ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                          @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                          @ T ) ) )
                @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_predi'.mono
thf(fact_6076_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys ) @ N2 )
        = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6077_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr316670251186196177_VEBTi @ ( produc316462671093861988_VEBTi @ Xs2 @ Ys ) @ N2 )
        = ( produc6084888613844515218_VEBTi @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6078_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr8725177398587324397T_VEBT @ ( produc1285381384045549624T_VEBT @ Xs2 @ Ys ) @ N2 )
        = ( produc7053807326796202854T_VEBT @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6079_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,Ys: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_s7982070591426661849_VEBTi @ Ys ) ) )
     => ( ( nth_Pr6329974346453275474_VEBTi @ ( produc194614972289024177_VEBTi @ Xs2 @ Ys ) @ N2 )
        = ( produc436343169921013763_VEBTi @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) @ ( nth_VEBT_VEBTi @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s7982070591426661849_VEBTi @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6080_product__nth,axiom,
    ! [N2: nat,Xs2: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N2 )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6081_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_real] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr6842391030413306568T_real @ ( produc4908677263432625371T_real @ Xs2 @ Ys ) @ N2 )
        = ( produc8117437818029410057T_real @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6082_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,Ys: list_real] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_real @ Ys ) ) )
     => ( ( nth_Pr3433448822664029129i_real @ ( produc5476717833281694120i_real @ Xs2 @ Ys ) @ N2 )
        = ( produc8457151488442208762i_real @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_real @ Ys ) ) ) @ ( nth_real @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_real @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6083_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys ) @ N2 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6084_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBTi,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr3306050735993963089EBTi_o @ ( product_VEBT_VEBTi_o @ Xs2 @ Ys ) @ N2 )
        = ( produc8194178580519725514EBTi_o @ ( nth_VEBT_VEBTi @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6085_product__nth,axiom,
    ! [N2: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys ) @ N2 )
        = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_6086_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_T8145700208782473153_VEBTi,K: vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T1006145433769338483_VEBTi @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ K )
      = ( heap_T5998771940306268294_VEBTi @ F
        @ ^ [X: $o] : ( heap_T1006145433769338483_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6087_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T8145700208782473153_VEBTi,K: vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T1006145433769338483_VEBTi @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ K )
      = ( heap_T844888390831797134_VEBTi @ F
        @ ^ [X: nat] : ( heap_T1006145433769338483_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6088_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T8145700208782473153_VEBTi,K: vEBT_VEBTi > heap_Time_Heap_o] :
      ( ( heap_T3040810144269856602EBTi_o @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ K )
      = ( heap_Time_bind_nat_o @ F
        @ ^ [X: nat] : ( heap_T3040810144269856602EBTi_o @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6089_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_o,K: $o > heap_Time_Heap_o] :
      ( ( heap_Time_bind_o_o @ ( heap_Time_bind_nat_o @ F @ G ) @ K )
      = ( heap_Time_bind_nat_o @ F
        @ ^ [X: nat] : ( heap_Time_bind_o_o @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6090_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_Time_Heap_o,K: $o > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T5998771940306268294_VEBTi @ ( heap_Time_bind_o_o @ F @ G ) @ K )
      = ( heap_T5998771940306268294_VEBTi @ F
        @ ^ [X: $o] : ( heap_T5998771940306268294_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6091_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_o,K: $o > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T5998771940306268294_VEBTi @ ( heap_Time_bind_nat_o @ F @ G ) @ K )
      = ( heap_T844888390831797134_VEBTi @ F
        @ ^ [X: nat] : ( heap_T5998771940306268294_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6092_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_Time_Heap_nat,K: nat > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T844888390831797134_VEBTi @ ( heap_Time_bind_o_nat @ F @ G ) @ K )
      = ( heap_T5998771940306268294_VEBTi @ F
        @ ^ [X: $o] : ( heap_T844888390831797134_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6093_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_nat,K: nat > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T844888390831797134_VEBTi @ ( heap_T7049098217575491753at_nat @ F @ G ) @ K )
      = ( heap_T844888390831797134_VEBTi @ F
        @ ^ [X: nat] : ( heap_T844888390831797134_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6094_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_nat,K: nat > heap_Time_Heap_o] :
      ( ( heap_Time_bind_nat_o @ ( heap_T7049098217575491753at_nat @ F @ G ) @ K )
      = ( heap_Time_bind_nat_o @ F
        @ ^ [X: nat] : ( heap_Time_bind_nat_o @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6095_Heap__Time__Monad_Obind__bind,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,G: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi,K: vEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( heap_T1006145433769338483_VEBTi @ ( heap_T5877712393672139267_VEBTi @ F @ G ) @ K )
      = ( heap_T5877712393672139267_VEBTi @ F
        @ ^ [X: list_VEBT_VEBTi] : ( heap_T1006145433769338483_VEBTi @ ( G @ X ) @ K ) ) ) ).

% Heap_Time_Monad.bind_bind
thf(fact_6096_curry__conv,axiom,
    ( produc1114182431767986483on_nat
    = ( ^ [F4: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat,A2: vEBT_VEBT,B2: vEBT_VEBTi] : ( F4 @ ( produc6084888613844515218_VEBTi @ A2 @ B2 ) ) ) ) ).

% curry_conv
thf(fact_6097_curry__conv,axiom,
    ( produc2663629013181010545Heap_o
    = ( ^ [F4: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o,A2: vEBT_VEBT,B2: vEBT_VEBTi] : ( F4 @ ( produc6084888613844515218_VEBTi @ A2 @ B2 ) ) ) ) ).

% curry_conv
thf(fact_6098_curry__conv,axiom,
    ( produc2164094337957399884_VEBTi
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi,A2: produc3625547720036274456_VEBTi,B2: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A2 @ B2 ) ) ) ) ).

% curry_conv
thf(fact_6099_curry__conv,axiom,
    ( produc1757988346207259447on_nat
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat,A2: produc3625547720036274456_VEBTi,B2: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A2 @ B2 ) ) ) ) ).

% curry_conv
thf(fact_6100_curry__conv,axiom,
    ( produc8381543706267210711Heap_o
    = ( ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o,A2: produc3625547720036274456_VEBTi,B2: nat] : ( F4 @ ( produc1853644041309157249Ti_nat @ A2 @ B2 ) ) ) ) ).

% curry_conv
thf(fact_6101_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_6102_curry__case__prod,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1114182431767986483on_nat @ ( produc1061038227461121684on_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6103_curry__case__prod,axiom,
    ! [F: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o] :
      ( ( produc2663629013181010545Heap_o @ ( produc2327743382103342416Heap_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6104_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
      ( ( produc2164094337957399884_VEBTi @ ( produc2943724498215716011_VEBTi @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6105_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1757988346207259447on_nat @ ( produc183673358652719894on_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6106_curry__case__prod,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o] :
      ( ( produc8381543706267210711Heap_o @ ( produc5872130906356439992Heap_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6107_curry__case__prod,axiom,
    ! [F: nat > nat > product_prod_nat_nat] :
      ( ( produc6629854527392350932at_nat @ ( produc2626176000494625587at_nat @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6108_curry__case__prod,axiom,
    ! [F: nat > nat > $o] :
      ( ( produc1310100445399344235_nat_o @ ( produc6081775807080527818_nat_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6109_curry__case__prod,axiom,
    ! [F: int > int > product_prod_int_int] :
      ( ( produc8249235968001453780nt_int @ ( produc4245557441103728435nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6110_curry__case__prod,axiom,
    ! [F: int > int > $o] :
      ( ( produc175634133007206835_int_o @ ( produc4947309494688390418_int_o @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6111_curry__case__prod,axiom,
    ! [F: int > int > int] :
      ( ( produc1016772743285680337nt_int @ ( produc8211389475949308722nt_int @ F ) )
      = F ) ).

% curry_case_prod
thf(fact_6112_case__prod__curry,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat] :
      ( ( produc1061038227461121684on_nat @ ( produc1114182431767986483on_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6113_case__prod__curry,axiom,
    ! [F: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o] :
      ( ( produc2327743382103342416Heap_o @ ( produc2663629013181010545Heap_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6114_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
      ( ( produc2943724498215716011_VEBTi @ ( produc2164094337957399884_VEBTi @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6115_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
      ( ( produc183673358652719894on_nat @ ( produc1757988346207259447on_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6116_case__prod__curry,axiom,
    ! [F: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
      ( ( produc5872130906356439992Heap_o @ ( produc8381543706267210711Heap_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6117_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ ( produc6629854527392350932at_nat @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6118_case__prod__curry,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o @ ( produc1310100445399344235_nat_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6119_case__prod__curry,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ ( produc8249235968001453780nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6120_case__prod__curry,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o @ ( produc175634133007206835_int_o @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6121_case__prod__curry,axiom,
    ! [F: product_prod_int_int > int] :
      ( ( produc8211389475949308722nt_int @ ( produc1016772743285680337nt_int @ F ) )
      = F ) ).

% case_prod_curry
thf(fact_6122_tanh__real__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( tanh_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% tanh_real_zero_iff
thf(fact_6123_tanh__real__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% tanh_real_less_iff
thf(fact_6124_tanh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6125_tanh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_pos_iff
thf(fact_6126_tanh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% tanh_real_nonneg_iff
thf(fact_6127_tanh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_6128_length__product,axiom,
    ! [Xs2: list_real,Ys: list_real] :
      ( ( size_s3932428310213730859l_real @ ( product_real_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6129_length__product,axiom,
    ! [Xs2: list_real,Ys: list_o] :
      ( ( size_s987546567493390085real_o @ ( product_real_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6130_length__product,axiom,
    ! [Xs2: list_real,Ys: list_nat] :
      ( ( size_s1877336372972134351al_nat @ ( product_real_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6131_length__product,axiom,
    ! [Xs2: list_real,Ys: list_int] :
      ( ( size_s8610625264895183403al_int @ ( product_real_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_real @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_6132_length__product,axiom,
    ! [Xs2: list_o,Ys: list_real] :
      ( ( size_s2624279037499656343o_real @ ( product_o_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6133_length__product,axiom,
    ! [Xs2: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6134_length__product,axiom,
    ! [Xs2: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_6135_length__product,axiom,
    ! [Xs2: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_6136_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_real] :
      ( ( size_s7910714270633306959t_real @ ( product_nat_real @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_real @ Ys ) ) ) ).

% length_product
thf(fact_6137_length__product,axiom,
    ! [Xs2: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_6138_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_nat,C4: nat > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple3826860765959394442ap_nat @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T645984214031783516rd_nat @ B6 )
     => ( ! [Y3: nat] : ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi @ ( C4 @ Y3 ) )
       => ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_T844888390831797134_VEBTi @ ( B6 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6139_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi,C4: vEBT_VEBTi > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi @ B6 )
     => ( ! [Y3: vEBT_VEBTi] : ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi @ ( C4 @ Y3 ) )
       => ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_T1006145433769338483_VEBTi @ ( B6 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6140_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_nat,C4: nat > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o] :
      ( ( comple3826860765959394442ap_nat @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T645984214031783516rd_nat @ B6 )
     => ( ! [Y3: nat] : ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o @ ( C4 @ Y3 ) )
       => ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_Time_bind_nat_o @ ( B6 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6141_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi,C4: vEBT_VEBTi > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o] :
      ( ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi @ B6 )
     => ( ! [Y3: vEBT_VEBTi] : ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o @ ( C4 @ Y3 ) )
       => ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_T3040810144269856602EBTi_o @ ( B6 @ F4 )
              @ ^ [Y: vEBT_VEBTi] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6142_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o,C4: $o > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o @ B6 )
     => ( ! [Y3: $o] : ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi @ ( C4 @ Y3 ) )
       => ( comple5335682857743707887_VEBTi @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_T5998771940306268294_VEBTi @ ( B6 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6143_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o,C4: $o > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > heap_Time_Heap_o] :
      ( ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o @ B6 )
     => ( ! [Y3: $o] : ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o @ ( C4 @ Y3 ) )
       => ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
              ( heap_Time_bind_o_o @ ( B6 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6144_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_nat,C4: nat > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o] :
      ( ( comple1015018851985181128ap_nat @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T645984214031783516rd_nat @ B6 )
     => ( ! [Y3: nat] : ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o @ ( C4 @ Y3 ) )
       => ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
              ( heap_Time_bind_nat_o @ ( B6 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6145_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_nat,C4: nat > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple1015018851985181128ap_nat @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T645984214031783516rd_nat @ B6 )
     => ( ! [Y3: nat] : ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi @ ( C4 @ Y3 ) )
       => ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
              ( heap_T844888390831797134_VEBTi @ ( B6 @ F4 )
              @ ^ [Y: nat] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6146_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o,C4: $o > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o] :
      ( ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o @ B6 )
     => ( ! [Y3: $o] : ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o @ ( C4 @ Y3 ) )
       => ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
              ( heap_Time_bind_o_o @ ( B6 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6147_Heap__Time__Monad_Obind__mono,axiom,
    ! [B6: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_Time_Heap_o,C4: $o > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > heap_T8145700208782473153_VEBTi] :
      ( ( comple6677746081827660726Heap_o @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_Time_Heap_ord_o @ B6 )
     => ( ! [Y3: $o] : ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi @ ( C4 @ Y3 ) )
       => ( comple2969382418784824877_VEBTi @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
              ( heap_T5998771940306268294_VEBTi @ ( B6 @ F4 )
              @ ^ [Y: $o] : ( C4 @ Y @ F4 ) ) ) ) ) ).

% Heap_Time_Monad.bind_mono
thf(fact_6148_heap_Oconst__mono,axiom,
    ! [Ord: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > $o,C: heap_T2636463487746394924on_nat] :
      ( comple4655144769394346904on_nat @ Ord @ heap_T7875578835903804844on_nat
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] : C ) ).

% heap.const_mono
thf(fact_6149_heap_Oconst__mono,axiom,
    ! [Ord: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > $o,C: heap_Time_Heap_o] :
      ( comple4217288648910406772Heap_o @ Ord @ heap_Time_Heap_ord_o
      @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] : C ) ).

% heap.const_mono
thf(fact_6150_heap_Oconst__mono,axiom,
    ! [Ord: ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > $o,C: heap_T8145700208782473153_VEBTi] :
      ( comple5606513277678308283_VEBTi @ Ord @ heap_T7173139186834293313_VEBTi
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] : C ) ).

% heap.const_mono
thf(fact_6151_heap_Oconst__mono,axiom,
    ! [Ord: ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > $o,C: heap_T2636463487746394924on_nat] :
      ( comple6977564771798581627on_nat @ Ord @ heap_T7875578835903804844on_nat
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] : C ) ).

% heap.const_mono
thf(fact_6152_heap_Oconst__mono,axiom,
    ! [Ord: ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > $o,C: heap_Time_Heap_o] :
      ( comple6074371103668693207Heap_o @ Ord @ heap_Time_Heap_ord_o
      @ ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] : C ) ).

% heap.const_mono
thf(fact_6153_curry__K,axiom,
    ! [C: nat > heap_T2636463487746394924on_nat] :
      ( ( produc1114182431767986483on_nat
        @ ^ [X: produc3625547720036274456_VEBTi] : C )
      = ( ^ [X: vEBT_VEBT,Y: vEBT_VEBTi] : C ) ) ).

% curry_K
thf(fact_6154_curry__K,axiom,
    ! [C: nat > heap_Time_Heap_o] :
      ( ( produc2663629013181010545Heap_o
        @ ^ [X: produc3625547720036274456_VEBTi] : C )
      = ( ^ [X: vEBT_VEBT,Y: vEBT_VEBTi] : C ) ) ).

% curry_K
thf(fact_6155_curry__K,axiom,
    ! [C: heap_T8145700208782473153_VEBTi] :
      ( ( produc2164094337957399884_VEBTi
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6156_curry__K,axiom,
    ! [C: heap_T2636463487746394924on_nat] :
      ( ( produc1757988346207259447on_nat
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6157_curry__K,axiom,
    ! [C: heap_Time_Heap_o] :
      ( ( produc8381543706267210711Heap_o
        @ ^ [X: produc3960855945107176009Ti_nat] : C )
      = ( ^ [X: produc3625547720036274456_VEBTi,Y: nat] : C ) ) ).

% curry_K
thf(fact_6158_tanh__real__lt__1,axiom,
    ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6159_curry__def,axiom,
    ( produc1114182431767986483on_nat
    = ( ^ [C3: produc3625547720036274456_VEBTi > nat > heap_T2636463487746394924on_nat,X: vEBT_VEBT,Y: vEBT_VEBTi] : ( C3 @ ( produc6084888613844515218_VEBTi @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_6160_curry__def,axiom,
    ( produc2663629013181010545Heap_o
    = ( ^ [C3: produc3625547720036274456_VEBTi > nat > heap_Time_Heap_o,X: vEBT_VEBT,Y: vEBT_VEBTi] : ( C3 @ ( produc6084888613844515218_VEBTi @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_6161_curry__def,axiom,
    ( produc2164094337957399884_VEBTi
    = ( ^ [C3: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi,X: produc3625547720036274456_VEBTi,Y: nat] : ( C3 @ ( produc1853644041309157249Ti_nat @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_6162_curry__def,axiom,
    ( produc1757988346207259447on_nat
    = ( ^ [C3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat,X: produc3625547720036274456_VEBTi,Y: nat] : ( C3 @ ( produc1853644041309157249Ti_nat @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_6163_curry__def,axiom,
    ( produc8381543706267210711Heap_o
    = ( ^ [C3: produc3960855945107176009Ti_nat > heap_Time_Heap_o,X: produc3625547720036274456_VEBTi,Y: nat] : ( C3 @ ( produc1853644041309157249Ti_nat @ X @ Y ) ) ) ) ).

% curry_def
thf(fact_6164_fold__if__return,axiom,
    ! [B3: $o,C: option_nat,D: option_nat] :
      ( ( B3
       => ( ( heap_T3487192422709364219on_nat @ C )
          = ( heap_T3487192422709364219on_nat @ ( if_option_nat @ B3 @ C @ D ) ) ) )
      & ( ~ B3
       => ( ( heap_T3487192422709364219on_nat @ D )
          = ( heap_T3487192422709364219on_nat @ ( if_option_nat @ B3 @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6165_fold__if__return,axiom,
    ! [B3: $o,C: nat,D: nat] :
      ( ( B3
       => ( ( heap_Time_return_nat @ C )
          = ( heap_Time_return_nat @ ( if_nat @ B3 @ C @ D ) ) ) )
      & ( ~ B3
       => ( ( heap_Time_return_nat @ D )
          = ( heap_Time_return_nat @ ( if_nat @ B3 @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6166_fold__if__return,axiom,
    ! [B3: $o,C: $o,D: $o] :
      ( ( B3
       => ( ( heap_Time_return_o @ C )
          = ( heap_Time_return_o
            @ ( ( B3
               => C )
              & ( ~ B3
               => D ) ) ) ) )
      & ( ~ B3
       => ( ( heap_Time_return_o @ D )
          = ( heap_Time_return_o
            @ ( ( B3
               => C )
              & ( ~ B3
               => D ) ) ) ) ) ) ).

% fold_if_return
thf(fact_6167_fold__if__return,axiom,
    ! [B3: $o,C: vEBT_VEBTi,D: vEBT_VEBTi] :
      ( ( B3
       => ( ( heap_T3630416162098727440_VEBTi @ C )
          = ( heap_T3630416162098727440_VEBTi @ ( if_VEBT_VEBTi @ B3 @ C @ D ) ) ) )
      & ( ~ B3
       => ( ( heap_T3630416162098727440_VEBTi @ D )
          = ( heap_T3630416162098727440_VEBTi @ ( if_VEBT_VEBTi @ B3 @ C @ D ) ) ) ) ) ).

% fold_if_return
thf(fact_6168_effect__deterministic_I3_J,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: option_nat,N2: nat,H7: heap_e7401611519738050253t_unit,B3: option_nat,N6: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T306965388786959644on_nat @ F @ H2 @ H7 @ B3 @ N6 )
       => ( N2 = N6 ) ) ) ).

% effect_deterministic(3)
thf(fact_6169_effect__deterministic_I3_J,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: $o,N2: nat,H7: heap_e7401611519738050253t_unit,B3: $o,N6: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_Time_effect_o @ F @ H2 @ H7 @ B3 @ N6 )
       => ( N2 = N6 ) ) ) ).

% effect_deterministic(3)
thf(fact_6170_effect__deterministic_I3_J,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: vEBT_VEBTi,N2: nat,H7: heap_e7401611519738050253t_unit,B3: vEBT_VEBTi,N6: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H7 @ B3 @ N6 )
       => ( N2 = N6 ) ) ) ).

% effect_deterministic(3)
thf(fact_6171_effect__deterministic_I2_J,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: option_nat,N2: nat,H7: heap_e7401611519738050253t_unit,B3: option_nat,N6: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T306965388786959644on_nat @ F @ H2 @ H7 @ B3 @ N6 )
       => ( H3 = H7 ) ) ) ).

% effect_deterministic(2)
thf(fact_6172_effect__deterministic_I2_J,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: $o,N2: nat,H7: heap_e7401611519738050253t_unit,B3: $o,N6: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_Time_effect_o @ F @ H2 @ H7 @ B3 @ N6 )
       => ( H3 = H7 ) ) ) ).

% effect_deterministic(2)
thf(fact_6173_effect__deterministic_I2_J,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: vEBT_VEBTi,N2: nat,H7: heap_e7401611519738050253t_unit,B3: vEBT_VEBTi,N6: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H7 @ B3 @ N6 )
       => ( H3 = H7 ) ) ) ).

% effect_deterministic(2)
thf(fact_6174_effect__deterministic_I1_J,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: option_nat,N2: nat,H7: heap_e7401611519738050253t_unit,B3: option_nat,N6: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T306965388786959644on_nat @ F @ H2 @ H7 @ B3 @ N6 )
       => ( A3 = B3 ) ) ) ).

% effect_deterministic(1)
thf(fact_6175_effect__deterministic_I1_J,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: $o,N2: nat,H7: heap_e7401611519738050253t_unit,B3: $o,N6: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_Time_effect_o @ F @ H2 @ H7 @ B3 @ N6 )
       => ( A3 = B3 ) ) ) ).

% effect_deterministic(1)
thf(fact_6176_effect__deterministic_I1_J,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,A3: vEBT_VEBTi,N2: nat,H7: heap_e7401611519738050253t_unit,B3: vEBT_VEBTi,N6: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H3 @ A3 @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H7 @ B3 @ N6 )
       => ( A3 = B3 ) ) ) ).

% effect_deterministic(1)
thf(fact_6177_effect__ifE,axiom,
    ! [C: $o,T2: heap_T2636463487746394924on_nat,E2: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( if_Hea5867803462524415986on_nat @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 )
     => ( ( C
         => ~ ( heap_T306965388786959644on_nat @ T2 @ H2 @ H3 @ R @ N2 ) )
       => ~ ( ~ C
           => ~ ( heap_T306965388786959644on_nat @ E2 @ H2 @ H3 @ R @ N2 ) ) ) ) ).

% effect_ifE
thf(fact_6178_effect__ifE,axiom,
    ! [C: $o,T2: heap_Time_Heap_o,E2: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( if_Heap_Time_Heap_o @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 )
     => ( ( C
         => ~ ( heap_Time_effect_o @ T2 @ H2 @ H3 @ R @ N2 ) )
       => ~ ( ~ C
           => ~ ( heap_Time_effect_o @ E2 @ H2 @ H3 @ R @ N2 ) ) ) ) ).

% effect_ifE
thf(fact_6179_effect__ifE,axiom,
    ! [C: $o,T2: heap_T8145700208782473153_VEBTi,E2: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 )
     => ( ( C
         => ~ ( heap_T2071195472996403633_VEBTi @ T2 @ H2 @ H3 @ R @ N2 ) )
       => ~ ( ~ C
           => ~ ( heap_T2071195472996403633_VEBTi @ E2 @ H2 @ H3 @ R @ N2 ) ) ) ) ).

% effect_ifE
thf(fact_6180_effect__ifI,axiom,
    ! [C: $o,T2: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat,E2: heap_T2636463487746394924on_nat] :
      ( ( C
       => ( heap_T306965388786959644on_nat @ T2 @ H2 @ H3 @ R @ N2 ) )
     => ( ( ~ C
         => ( heap_T306965388786959644on_nat @ E2 @ H2 @ H3 @ R @ N2 ) )
       => ( heap_T306965388786959644on_nat @ ( if_Hea5867803462524415986on_nat @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 ) ) ) ).

% effect_ifI
thf(fact_6181_effect__ifI,axiom,
    ! [C: $o,T2: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat,E2: heap_Time_Heap_o] :
      ( ( C
       => ( heap_Time_effect_o @ T2 @ H2 @ H3 @ R @ N2 ) )
     => ( ( ~ C
         => ( heap_Time_effect_o @ E2 @ H2 @ H3 @ R @ N2 ) )
       => ( heap_Time_effect_o @ ( if_Heap_Time_Heap_o @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 ) ) ) ).

% effect_ifI
thf(fact_6182_effect__ifI,axiom,
    ! [C: $o,T2: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat,E2: heap_T8145700208782473153_VEBTi] :
      ( ( C
       => ( heap_T2071195472996403633_VEBTi @ T2 @ H2 @ H3 @ R @ N2 ) )
     => ( ( ~ C
         => ( heap_T2071195472996403633_VEBTi @ E2 @ H2 @ H3 @ R @ N2 ) )
       => ( heap_T2071195472996403633_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ C @ T2 @ E2 ) @ H2 @ H3 @ R @ N2 ) ) ) ).

% effect_ifI
thf(fact_6183_distrib__if__bind,axiom,
    ! [B3: $o,C: heap_T4980287057938770641_VEBTi,D: heap_T4980287057938770641_VEBTi,F: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi] :
      ( ( B3
       => ( ( heap_T5877712393672139267_VEBTi @ ( if_Hea811341299636385687_VEBTi @ B3 @ C @ D ) @ F )
          = ( heap_T5877712393672139267_VEBTi @ C @ F ) ) )
      & ( ~ B3
       => ( ( heap_T5877712393672139267_VEBTi @ ( if_Hea811341299636385687_VEBTi @ B3 @ C @ D ) @ F )
          = ( heap_T5877712393672139267_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6184_distrib__if__bind,axiom,
    ! [B3: $o,C: heap_Time_Heap_o,D: heap_Time_Heap_o,F: $o > heap_T8145700208782473153_VEBTi] :
      ( ( B3
       => ( ( heap_T5998771940306268294_VEBTi @ ( if_Heap_Time_Heap_o @ B3 @ C @ D ) @ F )
          = ( heap_T5998771940306268294_VEBTi @ C @ F ) ) )
      & ( ~ B3
       => ( ( heap_T5998771940306268294_VEBTi @ ( if_Heap_Time_Heap_o @ B3 @ C @ D ) @ F )
          = ( heap_T5998771940306268294_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6185_distrib__if__bind,axiom,
    ! [B3: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_T8145700208782473153_VEBTi] :
      ( ( B3
       => ( ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_T844888390831797134_VEBTi @ C @ F ) ) )
      & ( ~ B3
       => ( ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_T844888390831797134_VEBTi @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6186_distrib__if__bind,axiom,
    ! [B3: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_T2636463487746394924on_nat] :
      ( ( B3
       => ( ( heap_T8222160169144143993on_nat @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_T8222160169144143993on_nat @ C @ F ) ) )
      & ( ~ B3
       => ( ( heap_T8222160169144143993on_nat @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_T8222160169144143993on_nat @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6187_distrib__if__bind,axiom,
    ! [B3: $o,C: heap_Time_Heap_nat,D: heap_Time_Heap_nat,F: nat > heap_Time_Heap_o] :
      ( ( B3
       => ( ( heap_Time_bind_nat_o @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_Time_bind_nat_o @ C @ F ) ) )
      & ( ~ B3
       => ( ( heap_Time_bind_nat_o @ ( if_Hea2662716070787841314ap_nat @ B3 @ C @ D ) @ F )
          = ( heap_Time_bind_nat_o @ D @ F ) ) ) ) ).

% distrib_if_bind
thf(fact_6188_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,X2: option_nat] :
      ( ( H2 = H3 )
     => ( heap_T306965388786959644on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ H2 @ H3 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6189_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,X2: nat] :
      ( ( H2 = H3 )
     => ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X2 ) @ H2 @ H3 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6190_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,X2: $o] :
      ( ( H2 = H3 )
     => ( heap_Time_effect_o @ ( heap_Time_return_o @ X2 ) @ H2 @ H3 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6191_effect__returnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,X2: vEBT_VEBTi] :
      ( ( H2 = H3 )
     => ( heap_T2071195472996403633_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ H2 @ H3 @ X2 @ one_one_nat ) ) ).

% effect_returnI
thf(fact_6192_effect__returnE,axiom,
    ! [X2: option_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T3487192422709364219on_nat @ X2 ) @ H2 @ H3 @ R @ N2 )
     => ~ ( ( R = X2 )
         => ( ( H3 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6193_effect__returnE,axiom,
    ! [X2: nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: nat,N2: nat] :
      ( ( heap_Time_effect_nat @ ( heap_Time_return_nat @ X2 ) @ H2 @ H3 @ R @ N2 )
     => ~ ( ( R = X2 )
         => ( ( H3 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6194_effect__returnE,axiom,
    ! [X2: $o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_return_o @ X2 ) @ H2 @ H3 @ R @ N2 )
     => ~ ( ( R = X2 )
         => ( ( H3 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6195_effect__returnE,axiom,
    ! [X2: vEBT_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T3630416162098727440_VEBTi @ X2 ) @ H2 @ H3 @ R @ N2 )
     => ~ ( ( R = X2 )
         => ( ( H3 = H2 )
           => ( N2 != one_one_nat ) ) ) ) ).

% effect_returnE
thf(fact_6196_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_bind_nat_o @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6197_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_Time_bind_o_o @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( ( heap_Time_effect_o @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6198_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_T3040810144269856602EBTi_o @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6199_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6200_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( ( heap_Time_effect_o @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6201_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N2: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ ( heap_T1006145433769338483_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T2071195472996403633_VEBTi @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6202_effect__bindE,axiom,
    ! [F: heap_Time_Heap_nat,G: nat > heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T8222160169144143993on_nat @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: nat,N1: nat] :
            ( ( heap_Time_effect_nat @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T306965388786959644on_nat @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6203_effect__bindE,axiom,
    ! [F: heap_Time_Heap_o,G: $o > heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T6306279297776390513on_nat @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: $o,N1: nat] :
            ( ( heap_Time_effect_o @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T306965388786959644on_nat @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6204_effect__bindE,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,G: vEBT_VEBTi > heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: option_nat,N2: nat] :
      ( ( heap_T306965388786959644on_nat @ ( heap_T2868974464944644318on_nat @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N1: nat] :
            ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_T306965388786959644on_nat @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6205_effect__bindE,axiom,
    ! [F: heap_T2636463487746394924on_nat,G: option_nat > heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,R5: $o,N2: nat] :
      ( ( heap_Time_effect_o @ ( heap_T6471384023045698863_nat_o @ F @ G ) @ H2 @ H7 @ R5 @ N2 )
     => ~ ! [H8: heap_e7401611519738050253t_unit,R3: option_nat,N1: nat] :
            ( ( heap_T306965388786959644on_nat @ F @ H2 @ H8 @ R3 @ N1 )
           => ! [N22: nat] :
                ( ( heap_Time_effect_o @ ( G @ R3 ) @ H8 @ H7 @ R5 @ N22 )
               => ( N2
                 != ( plus_plus_nat @ N1 @ N22 ) ) ) ) ) ).

% effect_bindE
thf(fact_6206_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: nat,N2: nat,G: nat > heap_Time_Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N6: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_Time_effect_o @ ( heap_Time_bind_nat_o @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6207_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: nat,N2: nat,G: nat > heap_T8145700208782473153_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N6: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T844888390831797134_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6208_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat,G: $o > heap_Time_Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N6: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_Time_effect_o @ ( heap_Time_bind_o_o @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6209_effect__bindI,axiom,
    ! [F: heap_Time_Heap_o,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: $o,N2: nat,G: $o > heap_T8145700208782473153_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N6: nat] :
      ( ( heap_Time_effect_o @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5998771940306268294_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6210_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat,G: vEBT_VEBTi > heap_Time_Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N6: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_Time_effect_o @ ( heap_T3040810144269856602EBTi_o @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6211_effect__bindI,axiom,
    ! [F: heap_T8145700208782473153_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat,G: vEBT_VEBTi > heap_T8145700208782473153_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N6: nat] :
      ( ( heap_T2071195472996403633_VEBTi @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T1006145433769338483_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6212_effect__bindI,axiom,
    ! [F: heap_Time_Heap_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: nat,N2: nat,G: nat > heap_T2636463487746394924on_nat,H7: heap_e7401611519738050253t_unit,R5: option_nat,N6: nat] :
      ( ( heap_Time_effect_nat @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T306965388786959644on_nat @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T306965388786959644on_nat @ ( heap_T8222160169144143993on_nat @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6213_effect__bindI,axiom,
    ! [F: heap_T4980287057938770641_VEBTi,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: list_VEBT_VEBTi,N2: nat,G: list_VEBT_VEBTi > heap_T8145700208782473153_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N6: nat] :
      ( ( heap_T33481931004607297_VEBTi @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5877712393672139267_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6214_effect__bindI,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat,G: option_nat > heap_Time_Heap_o,H7: heap_e7401611519738050253t_unit,R5: $o,N6: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_Time_effect_o @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_Time_effect_o @ ( heap_T6471384023045698863_nat_o @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6215_effect__bindI,axiom,
    ! [F: heap_T2636463487746394924on_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: option_nat,N2: nat,G: option_nat > heap_T8145700208782473153_VEBTi,H7: heap_e7401611519738050253t_unit,R5: vEBT_VEBTi,N6: nat] :
      ( ( heap_T306965388786959644on_nat @ F @ H2 @ H3 @ R @ N2 )
     => ( ( heap_T2071195472996403633_VEBTi @ ( G @ R ) @ H3 @ H7 @ R5 @ N6 )
       => ( heap_T2071195472996403633_VEBTi @ ( heap_T5661892481228163294_VEBTi @ F @ G ) @ H2 @ H7 @ R5 @ ( plus_plus_nat @ N2 @ N6 ) ) ) ) ).

% effect_bindI
thf(fact_6216_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > $o] :
      ( ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
        @ ^ [Vebt_predi3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] : ( P @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_predi3: vEBT_VEBT,T: vEBT_VEBTi,Ti3: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F5: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBT,A2: vEBT_VEBTi,B2: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ X8 ) )
                        @ ^ [Uu: product_unit] :
                            ( produc2190226783763740553on_nat
                            @ ^ [Info3: option4927543243414619207at_nat] :
                                ( produc2606485630176857543on_nat
                                @ ^ [Deg3: nat] :
                                    ( produc6124225815318652422on_nat
                                    @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                        ( heap_T5999496708990702694on_nat
                                        @ ( refine_Imp_assert
                                          @ ( ( Info3 = Info2 )
                                            & ( Deg3 = Deg2 )
                                            & ( vEBT_is_Node @ X8 ) ) )
                                        @ ^ [Uv: product_unit] :
                                            ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ^ [Mima: product_prod_nat_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ B2 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                                                  @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                    @ ^ [L: nat] :
                                                        ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_T5999496708990702694on_nat
                                                            @ ( refine_Imp_assert
                                                              @ ( L
                                                                = ( vEBT_VEBT_low @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                            @ ^ [Uw: product_unit] :
                                                                ( heap_T5999496708990702694on_nat
                                                                @ ( refine_Imp_assert
                                                                  @ ( H
                                                                    = ( vEBT_VEBT_high @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Ux: product_unit] :
                                                                    ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                    @ ^ [Uy: product_unit] :
                                                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                                                            @ ^ [Minlow: option_nat] :
                                                                                ( heap_T5999496708990702694on_nat
                                                                                @ ( refine_Imp_assert
                                                                                  @ ( Minlow
                                                                                    = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                @ ^ [Uz: product_unit] :
                                                                                    ( if_Hea5867803462524415986on_nat
                                                                                    @ ( ( Minlow != none_nat )
                                                                                      & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( F5 @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                      @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                                    @ ( heap_T3669509953089699273on_nat @ ( F5 @ Summary3 @ Summary2 @ H )
                                                                                      @ ^ [Predsum: option_nat] :
                                                                                          ( heap_T5999496708990702694on_nat
                                                                                          @ ( refine_Imp_assert
                                                                                            @ ( ( Predsum = none_nat )
                                                                                              = ( ( vEBT_vebt_pred @ Summary3 @ H )
                                                                                                = none_nat ) ) )
                                                                                          @ ^ [Va: product_unit] :
                                                                                              ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ B2 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                              @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                                                                @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                    ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                                    @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info2 ) ) ) )
                            @ ( vEBT_c634343235235684882T_VEBT
                              @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                              @ ^ [C3: $o,D3: $o] : undefi7074909574607331924T_VEBT
                              @ X8 ) ) )
                    @ ^ [C3: $o,D3: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ ( if_Hea5867803462524415986on_nat @ D3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ C3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( B2 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ C3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                    @ A2 ) ) )
         => ( P @ vEBT_VEBT_vebt_predi ) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
thf(fact_6217_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > $o] :
      ( ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
        @ ^ [Vebt_succi3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] : ( P @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_succi3 ) ) ) )
     => ( ( P
          @ ^ [Vebt_succi3: vEBT_VEBT,T: vEBT_VEBTi,Ti3: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F5: vEBT_VEBT > vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBT,A2: vEBT_VEBTi,B2: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ X8 ) )
                        @ ^ [Uu: product_unit] :
                            ( produc2190226783763740553on_nat
                            @ ^ [Info3: option4927543243414619207at_nat] :
                                ( produc2606485630176857543on_nat
                                @ ^ [Deg3: nat] :
                                    ( produc6124225815318652422on_nat
                                    @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                        ( heap_T5999496708990702694on_nat
                                        @ ( refine_Imp_assert
                                          @ ( ( Info3 = Info2 )
                                            & ( Deg3 = Deg2 )
                                            & ( vEBT_is_Node @ X8 ) ) )
                                        @ ^ [Uv: product_unit] :
                                            ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                                            @ ^ [Mima: product_prod_nat_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ B2 @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                                                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ B2 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                      @ ^ [L: nat] :
                                                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                          @ ^ [H: nat] :
                                                              ( heap_T5999496708990702694on_nat
                                                              @ ( refine_Imp_assert
                                                                @ ( L
                                                                  = ( vEBT_VEBT_low @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                              @ ^ [Uw: product_unit] :
                                                                  ( heap_T5999496708990702694on_nat
                                                                  @ ( refine_Imp_assert
                                                                    @ ( H
                                                                      = ( vEBT_VEBT_high @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                  @ ^ [Ux: product_unit] :
                                                                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                      @ ^ [Uy: product_unit] :
                                                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                                                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                                                              @ ^ [Maxlow: option_nat] :
                                                                                  ( heap_T5999496708990702694on_nat
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( Maxlow
                                                                                      = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                  @ ^ [Uz: product_unit] :
                                                                                      ( if_Hea5867803462524415986on_nat
                                                                                      @ ( ( Maxlow != none_nat )
                                                                                        & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                                                      @ ( heap_T3669509953089699273on_nat @ ( F5 @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                                        @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                                                      @ ( heap_T3669509953089699273on_nat @ ( F5 @ Summary3 @ Summary2 @ H )
                                                                                        @ ^ [Succsum: option_nat] :
                                                                                            ( heap_T5999496708990702694on_nat
                                                                                            @ ( refine_Imp_assert
                                                                                              @ ( ( Succsum = none_nat )
                                                                                                = ( ( vEBT_vebt_succ @ Summary3 @ H )
                                                                                                  = none_nat ) ) )
                                                                                            @ ^ [Va: product_unit] :
                                                                                                ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                                      ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                                                                      @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                                            @ Info2 ) ) ) )
                            @ ( vEBT_c634343235235684882T_VEBT
                              @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                              @ ^ [C3: $o,D3: $o] : undefi7074909574607331924T_VEBT
                              @ X8 ) ) )
                    @ ^ [C3: $o,D3: $o] : ( if_Hea5867803462524415986on_nat @ ( B2 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ D3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                    @ A2 ) ) )
         => ( P @ vEBT_VEBT_vebt_succi ) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
thf(fact_6218_vebt__succi_Omono,axiom,
    ! [X2: produc3881548065746020326Ti_nat] :
      ( comple4655144769394346904on_nat @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
      @ ^ [Vebt_succi4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
          ( produc8911080112929139129on_nat
          @ ^ [T: vEBT_VEBTi,X: nat] :
              ( vEBT_c6250501799366334488on_nat
              @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ^ [Mima: product_prod_nat_nat] :
                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ X @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                        @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                          @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                            @ ^ [L: nat] :
                                ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [H: nat] :
                                    ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                    @ ^ [Aktnode: vEBT_VEBTi] :
                                        ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                        @ ^ [Maxlow: option_nat] :
                                            ( if_Hea5867803462524415986on_nat
                                            @ ( ( Maxlow != none_nat )
                                              & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                            @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_succi4 @ Aktnode @ L )
                                              @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                            @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_succi4 @ Summary2 @ H )
                                              @ ^ [Succsum: option_nat] :
                                                  ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                  @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                    @ ^ [Nextnode: vEBT_VEBTi] :
                                                        ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                        @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info2 )
              @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( X = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
              @ T )
          @ X2 ) ) ).

% vebt_succi.mono
thf(fact_6219_vebt__predi_Omono,axiom,
    ! [X2: produc3881548065746020326Ti_nat] :
      ( comple4655144769394346904on_nat @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
      @ ^ [Vebt_predi4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
          ( produc8911080112929139129on_nat
          @ ^ [T: vEBT_VEBTi,X: nat] :
              ( vEBT_c6250501799366334488on_nat
              @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                  ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                  @ ^ [Mima: product_prod_nat_nat] :
                      ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                      @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                        @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                          @ ^ [L: nat] :
                              ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                              @ ^ [H: nat] :
                                  ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                  @ ^ [Aktnode: vEBT_VEBTi] :
                                      ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                      @ ^ [Minlow: option_nat] :
                                          ( if_Hea5867803462524415986on_nat
                                          @ ( ( Minlow != none_nat )
                                            & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                          @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_predi4 @ Aktnode @ L )
                                            @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                          @ ( heap_T3669509953089699273on_nat @ ( produc1489253303066280154on_nat @ Vebt_predi4 @ Summary2 @ H )
                                            @ ^ [Predsum: option_nat] :
                                                ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ X ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                      ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                      @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                  @ Info2 )
              @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( X = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
              @ T )
          @ X2 ) ) ).

% vebt_predi.mono
thf(fact_6220_pred__subset__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S4: set_Pr8218934625190621173um_num] :
      ( ( ord_le6124364862034508274_num_o
        @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R4 )
        @ ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S4 ) )
      = ( ord_le880128212290418581um_num @ R4 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_6221_pred__subset__eq2,axiom,
    ! [R4: set_Pr563407847431865468T_VEBT,S4: set_Pr563407847431865468T_VEBT] :
      ( ( ord_le870442331779451499VEBT_o
        @ ^ [X: nat,Y: produc4813437837504472865T_VEBT] : ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) @ R4 )
        @ ^ [X: nat,Y: produc4813437837504472865T_VEBT] : ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) @ S4 ) )
      = ( ord_le6438908469242860764T_VEBT @ R4 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_6222_pred__subset__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S4: set_Pr6200539531224447659at_num] :
      ( ( ord_le3404735783095501756_num_o
        @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R4 )
        @ ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S4 ) )
      = ( ord_le8085105155179020875at_num @ R4 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_6223_pred__subset__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S4 ) )
      = ( ord_le3146513528884898305at_nat @ R4 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_6224_pred__subset__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S4: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R4 )
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S4 ) )
      = ( ord_le2843351958646193337nt_int @ R4 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_6225_ln__one__minus__pos__lower__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_6226_curryI,axiom,
    ! [F: product_prod_num_num > $o,A3: num,B3: num] :
      ( ( F @ ( product_Pair_num_num @ A3 @ B3 ) )
     => ( produc932273227547478743_num_o @ F @ A3 @ B3 ) ) ).

% curryI
thf(fact_6227_curryI,axiom,
    ! [F: produc8398139464844984134T_VEBT > $o,A3: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( F @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) )
     => ( produc1960320730580999890VEBT_o @ F @ A3 @ B3 ) ) ).

% curryI
thf(fact_6228_curryI,axiom,
    ! [F: product_prod_nat_num > $o,A3: nat,B3: num] :
      ( ( F @ ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( produc156083480235303841_num_o @ F @ A3 @ B3 ) ) ).

% curryI
thf(fact_6229_curryI,axiom,
    ! [F: product_prod_nat_nat > $o,A3: nat,B3: nat] :
      ( ( F @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( produc1310100445399344235_nat_o @ F @ A3 @ B3 ) ) ).

% curryI
thf(fact_6230_curryI,axiom,
    ! [F: product_prod_int_int > $o,A3: int,B3: int] :
      ( ( F @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( produc175634133007206835_int_o @ F @ A3 @ B3 ) ) ).

% curryI
thf(fact_6231__092_060open_062_092_060And_062xa_Atia_O_Arefines_A_Ivebt__predi_Atia_Axa_J_A_IHeap_OHeap_AMap_Oempty_J_092_060close_062,axiom,
    ! [Tia2: vEBT_VEBTi,Xa: nat] :
      ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Tia2 @ Xa )
      @ ( heap_T5286843759275942675on_nat
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) ) ).

% \<open>\<And>xa tia. refines (vebt_predi tia xa) (Heap.Heap Map.empty)\<close>
thf(fact_6232_add_Oinverse__neutral,axiom,
    ( ( uminus8244633308260627903l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% add.inverse_neutral
thf(fact_6233_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_6234_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_uint32 @ zero_zero_uint32 )
    = zero_zero_uint32 ) ).

% add.inverse_neutral
thf(fact_6235_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_6236_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_6237_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_6238_neg__0__equal__iff__equal,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( zero_z3563351764282998399l_num1
        = ( uminus8244633308260627903l_num1 @ A3 ) )
      = ( zero_z3563351764282998399l_num1 = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6239_neg__0__equal__iff__equal,axiom,
    ! [A3: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A3 ) )
      = ( zero_zero_complex = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6240_neg__0__equal__iff__equal,axiom,
    ! [A3: uint32] :
      ( ( zero_zero_uint32
        = ( uminus_uminus_uint32 @ A3 ) )
      = ( zero_zero_uint32 = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6241_neg__0__equal__iff__equal,axiom,
    ! [A3: real] :
      ( ( zero_zero_real
        = ( uminus_uminus_real @ A3 ) )
      = ( zero_zero_real = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6242_neg__0__equal__iff__equal,axiom,
    ! [A3: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A3 ) )
      = ( zero_zero_rat = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6243_neg__0__equal__iff__equal,axiom,
    ! [A3: int] :
      ( ( zero_zero_int
        = ( uminus_uminus_int @ A3 ) )
      = ( zero_zero_int = A3 ) ) ).

% neg_0_equal_iff_equal
thf(fact_6244_neg__equal__0__iff__equal,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( ( uminus8244633308260627903l_num1 @ A3 )
        = zero_z3563351764282998399l_num1 )
      = ( A3 = zero_z3563351764282998399l_num1 ) ) ).

% neg_equal_0_iff_equal
thf(fact_6245_neg__equal__0__iff__equal,axiom,
    ! [A3: complex] :
      ( ( ( uminus1482373934393186551omplex @ A3 )
        = zero_zero_complex )
      = ( A3 = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_6246_neg__equal__0__iff__equal,axiom,
    ! [A3: uint32] :
      ( ( ( uminus_uminus_uint32 @ A3 )
        = zero_zero_uint32 )
      = ( A3 = zero_zero_uint32 ) ) ).

% neg_equal_0_iff_equal
thf(fact_6247_neg__equal__0__iff__equal,axiom,
    ! [A3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% neg_equal_0_iff_equal
thf(fact_6248_neg__equal__0__iff__equal,axiom,
    ! [A3: rat] :
      ( ( ( uminus_uminus_rat @ A3 )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_6249_neg__equal__0__iff__equal,axiom,
    ! [A3: int] :
      ( ( ( uminus_uminus_int @ A3 )
        = zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
thf(fact_6250_equal__neg__zero,axiom,
    ! [A3: real] :
      ( ( A3
        = ( uminus_uminus_real @ A3 ) )
      = ( A3 = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_6251_equal__neg__zero,axiom,
    ! [A3: rat] :
      ( ( A3
        = ( uminus_uminus_rat @ A3 ) )
      = ( A3 = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_6252_equal__neg__zero,axiom,
    ! [A3: int] :
      ( ( A3
        = ( uminus_uminus_int @ A3 ) )
      = ( A3 = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_6253_neg__equal__zero,axiom,
    ! [A3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = A3 )
      = ( A3 = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_6254_neg__equal__zero,axiom,
    ! [A3: rat] :
      ( ( ( uminus_uminus_rat @ A3 )
        = A3 )
      = ( A3 = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_6255_neg__equal__zero,axiom,
    ! [A3: int] :
      ( ( ( uminus_uminus_int @ A3 )
        = A3 )
      = ( A3 = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_6256_neg__le__iff__le,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_6257_neg__le__iff__le,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_6258_neg__le__iff__le,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_eq_int @ A3 @ B3 ) ) ).

% neg_le_iff_le
thf(fact_6259_neg__less__iff__less,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_real @ A3 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_6260_neg__less__iff__less,axiom,
    ! [B3: rat,A3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_rat @ A3 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_6261_neg__less__iff__less,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_int @ A3 @ B3 ) ) ).

% neg_less_iff_less
thf(fact_6262_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_6263_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_6264_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_6265_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_6266_mult__minus__left,axiom,
    ! [A3: complex,B3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_6267_mult__minus__left,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ B3 )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_6268_mult__minus__left,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_6269_mult__minus__left,axiom,
    ! [A3: rat,B3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
      = ( uminus_uminus_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_6270_mult__minus__left,axiom,
    ! [A3: int,B3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B3 )
      = ( uminus_uminus_int @ ( times_times_int @ A3 @ B3 ) ) ) ).

% mult_minus_left
thf(fact_6271_minus__mult__minus,axiom,
    ! [A3: complex,B3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
      = ( times_times_complex @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_6272_minus__mult__minus,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ ( uminus_uminus_uint32 @ B3 ) )
      = ( times_times_uint32 @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_6273_minus__mult__minus,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
      = ( times_times_real @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_6274_minus__mult__minus,axiom,
    ! [A3: rat,B3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) )
      = ( times_times_rat @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_6275_minus__mult__minus,axiom,
    ! [A3: int,B3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
      = ( times_times_int @ A3 @ B3 ) ) ).

% minus_mult_minus
thf(fact_6276_mult__minus__right,axiom,
    ! [A3: complex,B3: complex] :
      ( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_6277_mult__minus__right,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( times_times_uint32 @ A3 @ ( uminus_uminus_uint32 @ B3 ) )
      = ( uminus_uminus_uint32 @ ( times_times_uint32 @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_6278_mult__minus__right,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) )
      = ( uminus_uminus_real @ ( times_times_real @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_6279_mult__minus__right,axiom,
    ! [A3: rat,B3: rat] :
      ( ( times_times_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_6280_mult__minus__right,axiom,
    ! [A3: int,B3: int] :
      ( ( times_times_int @ A3 @ ( uminus_uminus_int @ B3 ) )
      = ( uminus_uminus_int @ ( times_times_int @ A3 @ B3 ) ) ) ).

% mult_minus_right
thf(fact_6281_div__minus__minus,axiom,
    ! [A3: int,B3: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
      = ( divide_divide_int @ A3 @ B3 ) ) ).

% div_minus_minus
thf(fact_6282_dvd__minus__iff,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ X2 @ ( uminus8244633308260627903l_num1 @ Y2 ) )
      = ( dvd_dv6812691276156420380l_num1 @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6283_dvd__minus__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( dvd_dvd_Code_integer @ X2 @ ( uminus1351360451143612070nteger @ Y2 ) )
      = ( dvd_dvd_Code_integer @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6284_dvd__minus__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( dvd_dvd_complex @ X2 @ ( uminus1482373934393186551omplex @ Y2 ) )
      = ( dvd_dvd_complex @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6285_dvd__minus__iff,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( dvd_dvd_uint32 @ X2 @ ( uminus_uminus_uint32 @ Y2 ) )
      = ( dvd_dvd_uint32 @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6286_dvd__minus__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( dvd_dvd_real @ X2 @ ( uminus_uminus_real @ Y2 ) )
      = ( dvd_dvd_real @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6287_dvd__minus__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( dvd_dvd_rat @ X2 @ ( uminus_uminus_rat @ Y2 ) )
      = ( dvd_dvd_rat @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6288_dvd__minus__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( dvd_dvd_int @ X2 @ ( uminus_uminus_int @ Y2 ) )
      = ( dvd_dvd_int @ X2 @ Y2 ) ) ).

% dvd_minus_iff
thf(fact_6289_minus__dvd__iff,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( uminus8244633308260627903l_num1 @ X2 ) @ Y2 )
      = ( dvd_dv6812691276156420380l_num1 @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6290_minus__dvd__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X2 ) @ Y2 )
      = ( dvd_dvd_Code_integer @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6291_minus__dvd__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X2 ) @ Y2 )
      = ( dvd_dvd_complex @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6292_minus__dvd__iff,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( dvd_dvd_uint32 @ ( uminus_uminus_uint32 @ X2 ) @ Y2 )
      = ( dvd_dvd_uint32 @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6293_minus__dvd__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X2 ) @ Y2 )
      = ( dvd_dvd_real @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6294_minus__dvd__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X2 ) @ Y2 )
      = ( dvd_dvd_rat @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6295_minus__dvd__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X2 ) @ Y2 )
      = ( dvd_dvd_int @ X2 @ Y2 ) ) ).

% minus_dvd_iff
thf(fact_6296_mod__minus__minus,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( uminus1351360451143612070nteger @ B3 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) ) ).

% mod_minus_minus
thf(fact_6297_mod__minus__minus,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ ( uminus_uminus_int @ B3 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ).

% mod_minus_minus
thf(fact_6298_neg__0__le__iff__le,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% neg_0_le_iff_le
thf(fact_6299_neg__0__le__iff__le,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_6300_neg__0__le__iff__le,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).

% neg_0_le_iff_le
thf(fact_6301_neg__le__0__iff__le,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% neg_le_0_iff_le
thf(fact_6302_neg__le__0__iff__le,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% neg_le_0_iff_le
thf(fact_6303_neg__le__0__iff__le,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).

% neg_le_0_iff_le
thf(fact_6304_less__eq__neg__nonpos,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ).

% less_eq_neg_nonpos
thf(fact_6305_less__eq__neg__nonpos,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_6306_less__eq__neg__nonpos,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_eq_int @ A3 @ zero_zero_int ) ) ).

% less_eq_neg_nonpos
thf(fact_6307_neg__less__eq__nonneg,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ A3 )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% neg_less_eq_nonneg
thf(fact_6308_neg__less__eq__nonneg,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% neg_less_eq_nonneg
thf(fact_6309_neg__less__eq__nonneg,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ A3 )
      = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).

% neg_less_eq_nonneg
thf(fact_6310_less__neg__neg,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% less_neg_neg
thf(fact_6311_less__neg__neg,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_6312_less__neg__neg,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ A3 @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_int @ A3 @ zero_zero_int ) ) ).

% less_neg_neg
thf(fact_6313_neg__less__pos,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ A3 )
      = ( ord_less_real @ zero_zero_real @ A3 ) ) ).

% neg_less_pos
thf(fact_6314_neg__less__pos,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
      = ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).

% neg_less_pos
thf(fact_6315_neg__less__pos,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ A3 )
      = ( ord_less_int @ zero_zero_int @ A3 ) ) ).

% neg_less_pos
thf(fact_6316_neg__0__less__iff__less,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A3 ) )
      = ( ord_less_real @ A3 @ zero_zero_real ) ) ).

% neg_0_less_iff_less
thf(fact_6317_neg__0__less__iff__less,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A3 ) )
      = ( ord_less_rat @ A3 @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_6318_neg__0__less__iff__less,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A3 ) )
      = ( ord_less_int @ A3 @ zero_zero_int ) ) ).

% neg_0_less_iff_less
thf(fact_6319_neg__less__0__iff__less,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ zero_zero_real )
      = ( ord_less_real @ zero_zero_real @ A3 ) ) ).

% neg_less_0_iff_less
thf(fact_6320_neg__less__0__iff__less,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A3 ) ) ).

% neg_less_0_iff_less
thf(fact_6321_neg__less__0__iff__less,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ zero_zero_int )
      = ( ord_less_int @ zero_zero_int @ A3 ) ) ).

% neg_less_0_iff_less
thf(fact_6322_add_Oright__inverse,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ A3 @ ( uminus8244633308260627903l_num1 @ A3 ) )
      = zero_z3563351764282998399l_num1 ) ).

% add.right_inverse
thf(fact_6323_add_Oright__inverse,axiom,
    ! [A3: complex] :
      ( ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ A3 ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_6324_add_Oright__inverse,axiom,
    ! [A3: uint32] :
      ( ( plus_plus_uint32 @ A3 @ ( uminus_uminus_uint32 @ A3 ) )
      = zero_zero_uint32 ) ).

% add.right_inverse
thf(fact_6325_add_Oright__inverse,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ A3 @ ( uminus_uminus_real @ A3 ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_6326_add_Oright__inverse,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ A3 ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_6327_add_Oright__inverse,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ A3 @ ( uminus_uminus_int @ A3 ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_6328_ab__left__minus,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ A3 ) @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% ab_left_minus
thf(fact_6329_ab__left__minus,axiom,
    ! [A3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ A3 )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_6330_ab__left__minus,axiom,
    ! [A3: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ A3 )
      = zero_zero_uint32 ) ).

% ab_left_minus
thf(fact_6331_ab__left__minus,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_6332_ab__left__minus,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_6333_ab__left__minus,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_6334_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) ) )
      = ( uminus8244633308260627903l_num1 @ ( plus_p361126936061061375l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( numera7442385471795722001l_num1 @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6335_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6336_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) ) )
      = ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( numera9087168376688890119uint32 @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6337_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6338_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6339_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_6340_diff__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ zero_z3563351764282998399l_num1 @ A3 )
      = ( uminus8244633308260627903l_num1 @ A3 ) ) ).

% diff_0
thf(fact_6341_diff__0,axiom,
    ! [A3: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A3 )
      = ( uminus1482373934393186551omplex @ A3 ) ) ).

% diff_0
thf(fact_6342_diff__0,axiom,
    ! [A3: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ A3 )
      = ( uminus_uminus_uint32 @ A3 ) ) ).

% diff_0
thf(fact_6343_diff__0,axiom,
    ! [A3: real] :
      ( ( minus_minus_real @ zero_zero_real @ A3 )
      = ( uminus_uminus_real @ A3 ) ) ).

% diff_0
thf(fact_6344_diff__0,axiom,
    ! [A3: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A3 )
      = ( uminus_uminus_rat @ A3 ) ) ).

% diff_0
thf(fact_6345_diff__0,axiom,
    ! [A3: int] :
      ( ( minus_minus_int @ zero_zero_int @ A3 )
      = ( uminus_uminus_int @ A3 ) ) ).

% diff_0
thf(fact_6346_verit__minus__simplify_I3_J,axiom,
    ! [B3: word_N3645301735248828278l_num1] :
      ( ( minus_4019991460397169231l_num1 @ zero_z3563351764282998399l_num1 @ B3 )
      = ( uminus8244633308260627903l_num1 @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6347_verit__minus__simplify_I3_J,axiom,
    ! [B3: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B3 )
      = ( uminus1482373934393186551omplex @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6348_verit__minus__simplify_I3_J,axiom,
    ! [B3: uint32] :
      ( ( minus_minus_uint32 @ zero_zero_uint32 @ B3 )
      = ( uminus_uminus_uint32 @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6349_verit__minus__simplify_I3_J,axiom,
    ! [B3: real] :
      ( ( minus_minus_real @ zero_zero_real @ B3 )
      = ( uminus_uminus_real @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6350_verit__minus__simplify_I3_J,axiom,
    ! [B3: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B3 )
      = ( uminus_uminus_rat @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6351_verit__minus__simplify_I3_J,axiom,
    ! [B3: int] :
      ( ( minus_minus_int @ zero_zero_int @ B3 )
      = ( uminus_uminus_int @ B3 ) ) ).

% verit_minus_simplify(3)
thf(fact_6352_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_6353_mult__minus1,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ Z )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1
thf(fact_6354_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_6355_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_6356_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_6357_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_6358_mult__minus1__right,axiom,
    ! [Z: uint32] :
      ( ( times_times_uint32 @ Z @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
      = ( uminus_uminus_uint32 @ Z ) ) ).

% mult_minus1_right
thf(fact_6359_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_6360_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_6361_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_6362_divide__minus1,axiom,
    ! [X2: complex] :
      ( ( divide1717551699836669952omplex @ X2 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X2 ) ) ).

% divide_minus1
thf(fact_6363_divide__minus1,axiom,
    ! [X2: real] :
      ( ( divide_divide_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X2 ) ) ).

% divide_minus1
thf(fact_6364_divide__minus1,axiom,
    ! [X2: rat] :
      ( ( divide_divide_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X2 ) ) ).

% divide_minus1
thf(fact_6365_div__minus1__right,axiom,
    ! [A3: int] :
      ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A3 ) ) ).

% div_minus1_right
thf(fact_6366_minus__mod__self1,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B3 @ A3 ) @ B3 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).

% minus_mod_self1
thf(fact_6367_minus__mod__self1,axiom,
    ! [B3: int,A3: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B3 @ A3 ) @ B3 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).

% minus_mod_self1
thf(fact_6368_real__add__minus__iff,axiom,
    ! [X2: real,A3: real] :
      ( ( ( plus_plus_real @ X2 @ ( uminus_uminus_real @ A3 ) )
        = zero_zero_real )
      = ( X2 = A3 ) ) ).

% real_add_minus_iff
thf(fact_6369_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ one_on7727431528512463931l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% add_neg_numeral_special(8)
thf(fact_6370_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_6371_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(8)
thf(fact_6372_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_6373_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_6374_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_6375_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = zero_z3563351764282998399l_num1 ) ).

% add_neg_numeral_special(7)
thf(fact_6376_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_6377_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% add_neg_numeral_special(7)
thf(fact_6378_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_6379_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_6380_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_6381_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6382_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6383_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6384_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_6385_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6386_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6387_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6388_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_6389_diff__numeral__special_I12_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = zero_z3563351764282998399l_num1 ) ).

% diff_numeral_special(12)
thf(fact_6390_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_6391_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = zero_zero_uint32 ) ).

% diff_numeral_special(12)
thf(fact_6392_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_6393_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_6394_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_6395_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_6396_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_6397_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 ) )
      = one_one_uint32 ) ).

% minus_one_mult_self
thf(fact_6398_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_6399_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_6400_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_6401_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6402_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6403_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 ) @ ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6404_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6405_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6406_left__minus__one__mult__self,axiom,
    ! [N2: nat,A3: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A3 ) )
      = A3 ) ).

% left_minus_one_mult_self
thf(fact_6407_mod__minus1__right,axiom,
    ! [A3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_6408_mod__minus1__right,axiom,
    ! [A3: int] :
      ( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_6409_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_6410_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
          = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) ) )
      & ( ~ ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
          = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6411_max__number__of_I4_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_6412_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_6413_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_6414_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_6415_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_6416_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( numera7442385471795722001l_num1 @ V ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( numera7442385471795722001l_num1 @ V ) )
          = ( numera7442385471795722001l_num1 @ V ) ) )
      & ( ~ ( ord_le3335648743751981014l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( numera7442385471795722001l_num1 @ V ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) @ ( numera7442385471795722001l_num1 @ V ) )
          = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6417_max__number__of_I3_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( numera9087168376688890119uint32 @ V ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
       => ( ( ord_max_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) @ ( numera9087168376688890119uint32 @ V ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_6418_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_6419_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_6420_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_6421_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_6422_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_le3335648743751981014l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
          = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) ) )
      & ( ~ ( ord_le3335648743751981014l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
       => ( ( ord_ma8239519435860878689l_num1 @ ( numera7442385471795722001l_num1 @ U ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) )
          = ( numera7442385471795722001l_num1 @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6423_max__number__of_I2_J,axiom,
    ! [U: num,V: num] :
      ( ( ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) ) )
      & ( ~ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
       => ( ( ord_max_uint32 @ ( numera9087168376688890119uint32 @ U ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) )
          = ( numera9087168376688890119uint32 @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_6424_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_6425_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_6426_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_6427_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_6428_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_6429_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y2 ) )
      = ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6430_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6431_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y2 ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6432_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6433_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6434_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(168)
thf(fact_6435_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_6436_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_6437_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_4019991460397169231l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6438_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6439_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6440_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6441_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6442_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_6443_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6444_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6445_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N2 ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6446_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6447_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6448_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_6449_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y2 ) )
      = ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6450_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6451_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y2 ) )
      = ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6452_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6453_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6454_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y2 ) ) ).

% semiring_norm(172)
thf(fact_6455_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ V ) @ ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) @ Y2 ) )
      = ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6456_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y2 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6457_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ V ) @ ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) @ Y2 ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6458_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y2 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6459_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y2 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6460_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y2 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(171)
thf(fact_6461_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ V ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ Y2 ) )
      = ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6462_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y2 ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6463_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ V ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ Y2 ) )
      = ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6464_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y2 ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6465_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y2 ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6466_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y2: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y2 ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y2 ) ) ).

% semiring_norm(170)
thf(fact_6467_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) ) )
      = ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6468_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6469_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) ) )
      = ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6470_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6471_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6472_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_6473_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ ( numera7442385471795722001l_num1 @ N2 ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6474_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6475_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ ( numera9087168376688890119uint32 @ N2 ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6476_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6477_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6478_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_6479_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ M ) @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6480_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6481_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ M ) @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6482_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6483_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6484_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_6485_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6486_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6487_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_6488_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6489_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6490_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_6491_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_6492_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_6493_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_6494_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_6495_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_6496_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_6497_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_6498_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_6499_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: complex,W: num,A3: complex] :
      ( ( ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A3 )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A3 = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_6500_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A3 )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B3
            = ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_6501_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A3 )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B3
            = ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_6502_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A3: complex,B3: complex,W: num] :
      ( ( A3
        = ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B3 ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A3 = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_6503_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( A3
        = ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B3 ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_6504_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( A3
        = ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B3 ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_6505_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A3 )
      = ( ord_less_eq_real @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B3 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_6506_divide__le__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A3 )
      = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B3 ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_6507_le__divide__eq__numeral1_I2_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ord_less_eq_real @ A3 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B3 @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_6508_le__divide__eq__numeral1_I2_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( ord_less_eq_rat @ A3 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_6509_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: real,W: num,A3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A3 )
      = ( ord_less_real @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B3 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_6510_divide__less__eq__numeral1_I2_J,axiom,
    ! [B3: rat,W: num,A3: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A3 )
      = ( ord_less_rat @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B3 ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_6511_less__divide__eq__numeral1_I2_J,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ord_less_real @ A3 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B3 @ ( times_times_real @ A3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_6512_less__divide__eq__numeral1_I2_J,axiom,
    ! [A3: rat,B3: rat,W: num] :
      ( ( ord_less_rat @ A3 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B3 @ ( times_times_rat @ A3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_6513_power2__minus,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6514_power2__minus,axiom,
    ! [A3: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6515_power2__minus,axiom,
    ! [A3: uint32] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_uint32 @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6516_power2__minus,axiom,
    ! [A3: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6517_power2__minus,axiom,
    ! [A3: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6518_power2__minus,axiom,
    ! [A3: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_6519_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_6520_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_6521_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_6522_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_6523_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_6524_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_6525_diff__numeral__special_I11_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_6526_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_6527_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_6528_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_6529_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_6530_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_6531_diff__numeral__special_I10_J,axiom,
    ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ one_on7727431528512463931l_num1 )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_6532_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_6533_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ one_one_uint32 )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo1504961113040953224l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) )
    = one_on7727431528512463931l_num1 ) ).

% bits_minus_1_mod_2_eq
thf(fact_6541_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_6542_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) )
    = one_one_uint32 ) ).

% bits_minus_1_mod_2_eq
thf(fact_6543_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_6544_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6545_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6546_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_uint32 @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6547_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6548_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_rat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6549_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_6550_power__minus__odd,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6551_power__minus__odd,axiom,
    ! [N2: nat,A3: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6552_power__minus__odd,axiom,
    ! [N2: nat,A3: uint32] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 )
        = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6553_power__minus__odd,axiom,
    ! [N2: nat,A3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 )
        = ( uminus_uminus_real @ ( power_power_real @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6554_power__minus__odd,axiom,
    ! [N2: nat,A3: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 )
        = ( uminus_uminus_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6555_power__minus__odd,axiom,
    ! [N2: nat,A3: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 )
        = ( uminus_uminus_int @ ( power_power_int @ A3 @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_6556_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 )
        = ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6557_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 )
        = ( power_power_complex @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6558_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 )
        = ( power_power_uint32 @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6559_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 )
        = ( power_power_real @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6560_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 )
        = ( power_power_rat @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6561_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A3: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 )
        = ( power_power_int @ A3 @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_6562_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_4019991460397169231l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ M ) ) @ one_on7727431528512463931l_num1 )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6563_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_6564_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ M ) ) @ one_one_uint32 )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_6565_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_6566_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_6567_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_6568_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_4019991460397169231l_num1 @ one_on7727431528512463931l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ N2 ) ) )
      = ( numera7442385471795722001l_num1 @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6569_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6570_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_uint32 @ one_one_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ N2 ) ) )
      = ( numera9087168376688890119uint32 @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6571_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6572_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6573_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_6574_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6575_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6576_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 )
        = ( ring_1_of_int_real @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6577_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6578_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = ( ring_1_of_int_int @ Y2 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 )
        = Y2 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6579_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y2 )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6580_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y2 )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6581_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y2 )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6582_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y2 )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6583_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y2 )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6584_ceiling__le__neg__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_6585_ceiling__le__neg__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_6586_neg__numeral__less__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).

% neg_numeral_less_ceiling
thf(fact_6587_neg__numeral__less__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).

% neg_numeral_less_ceiling
thf(fact_6588_ceiling__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_6589_ceiling__less__zero,axiom,
    ! [X2: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_6590_zero__le__ceiling,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).

% zero_le_ceiling
thf(fact_6591_zero__le__ceiling,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).

% zero_le_ceiling
thf(fact_6592_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_6593_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_6594_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_uint32 ) ).

% power_minus1_even
thf(fact_6595_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_6596_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_6597_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_6598_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_6599_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_6600_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 )
        = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ).

% neg_one_odd_power
thf(fact_6601_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_6602_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_6603_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_6604_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_6605_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_6606_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 )
        = one_one_uint32 ) ) ).

% neg_one_even_power
thf(fact_6607_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_6608_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_6609_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_6610_ceiling__less__neg__numeral,axiom,
    ! [X2: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_6611_ceiling__less__neg__numeral,axiom,
    ! [X2: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_6612_neg__numeral__le__ceiling,axiom,
    ! [V: num,X2: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).

% neg_numeral_le_ceiling
thf(fact_6613_neg__numeral__le__ceiling,axiom,
    ! [V: num,X2: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).

% neg_numeral_le_ceiling
thf(fact_6614_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6615_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) @ ( ring_1_of_int_real @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6616_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6617_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ ( ring_1_of_int_int @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6618_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6619_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6620_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6621_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6622_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6623_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) @ ( ring_1_of_int_real @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6624_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6625_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ ( ring_1_of_int_int @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) @ A3 ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6626_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A3 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6627_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A3 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6628_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6629_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A3 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6630_Heap__lub__empty,axiom,
    ( ( heap_T7048022066654196708on_nat @ bot_bo8932748503833948152on_nat )
    = ( heap_T5286843759275942675on_nat
      @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) ) ).

% Heap_lub_empty
thf(fact_6631_Heap__lub__empty,axiom,
    ( ( heap_Time_Heap_lub_o @ bot_bo3236126332025433324Heap_o )
    = ( heap_Time_Heap_o2
      @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) ) ).

% Heap_lub_empty
thf(fact_6632_Heap__lub__empty,axiom,
    ( ( heap_T3112222404744780921_VEBTi @ bot_bo3125955617464001165_VEBTi )
    = ( heap_T1489671670754571048_VEBTi
      @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) ) ).

% Heap_lub_empty
thf(fact_6633_le__minus__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ B3 ) )
      = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).

% le_minus_iff
thf(fact_6634_le__minus__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
      = ( ord_less_eq_rat @ B3 @ ( uminus_uminus_rat @ A3 ) ) ) ).

% le_minus_iff
thf(fact_6635_le__minus__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ ( uminus_uminus_int @ B3 ) )
      = ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A3 ) ) ) ).

% le_minus_iff
thf(fact_6636_minus__le__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).

% minus_le_iff
thf(fact_6637_minus__le__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ A3 ) ) ).

% minus_le_iff
thf(fact_6638_minus__le__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A3 ) ) ).

% minus_le_iff
thf(fact_6639_le__imp__neg__le,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).

% le_imp_neg_le
thf(fact_6640_le__imp__neg__le,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).

% le_imp_neg_le
thf(fact_6641_le__imp__neg__le,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).

% le_imp_neg_le
thf(fact_6642_less__minus__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ B3 ) )
      = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A3 ) ) ) ).

% less_minus_iff
thf(fact_6643_less__minus__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
      = ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ A3 ) ) ) ).

% less_minus_iff
thf(fact_6644_less__minus__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ ( uminus_uminus_int @ B3 ) )
      = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A3 ) ) ) ).

% less_minus_iff
thf(fact_6645_minus__less__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A3 ) ) ).

% minus_less_iff
thf(fact_6646_minus__less__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ A3 ) ) ).

% minus_less_iff
thf(fact_6647_minus__less__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B3 )
      = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A3 ) ) ).

% minus_less_iff
thf(fact_6648_verit__negate__coefficient_I2_J,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_6649_verit__negate__coefficient_I2_J,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_6650_verit__negate__coefficient_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ B3 )
     => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_6651_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_6652_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_6653_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_6654_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_6655_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6656_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6657_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6658_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_6659_is__num__normalize_I8_J,axiom,
    ! [A3: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A3 @ B3 ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A3 ) ) ) ).

% is_num_normalize(8)
thf(fact_6660_is__num__normalize_I8_J,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( uminus_uminus_uint32 @ ( plus_plus_uint32 @ A3 @ B3 ) )
      = ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ B3 ) @ ( uminus_uminus_uint32 @ A3 ) ) ) ).

% is_num_normalize(8)
thf(fact_6661_is__num__normalize_I8_J,axiom,
    ! [A3: real,B3: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A3 @ B3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A3 ) ) ) ).

% is_num_normalize(8)
thf(fact_6662_is__num__normalize_I8_J,axiom,
    ! [A3: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A3 @ B3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A3 ) ) ) ).

% is_num_normalize(8)
thf(fact_6663_is__num__normalize_I8_J,axiom,
    ! [A3: int,B3: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A3 @ B3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A3 ) ) ) ).

% is_num_normalize(8)
thf(fact_6664_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_6665_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_6666_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_6667_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_6668_square__eq__iff,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ( times_times_complex @ A3 @ A3 )
        = ( times_times_complex @ B3 @ B3 ) )
      = ( ( A3 = B3 )
        | ( A3
          = ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% square_eq_iff
thf(fact_6669_square__eq__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( times_times_real @ A3 @ A3 )
        = ( times_times_real @ B3 @ B3 ) )
      = ( ( A3 = B3 )
        | ( A3
          = ( uminus_uminus_real @ B3 ) ) ) ) ).

% square_eq_iff
thf(fact_6670_square__eq__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( times_times_rat @ A3 @ A3 )
        = ( times_times_rat @ B3 @ B3 ) )
      = ( ( A3 = B3 )
        | ( A3
          = ( uminus_uminus_rat @ B3 ) ) ) ) ).

% square_eq_iff
thf(fact_6671_square__eq__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ( times_times_int @ A3 @ A3 )
        = ( times_times_int @ B3 @ B3 ) )
      = ( ( A3 = B3 )
        | ( A3
          = ( uminus_uminus_int @ B3 ) ) ) ) ).

% square_eq_iff
thf(fact_6672_minus__mult__commute,axiom,
    ! [A3: complex,B3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
      = ( times_times_complex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).

% minus_mult_commute
thf(fact_6673_minus__mult__commute,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ B3 )
      = ( times_times_uint32 @ A3 @ ( uminus_uminus_uint32 @ B3 ) ) ) ).

% minus_mult_commute
thf(fact_6674_minus__mult__commute,axiom,
    ! [A3: real,B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A3 ) @ B3 )
      = ( times_times_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).

% minus_mult_commute
thf(fact_6675_minus__mult__commute,axiom,
    ! [A3: rat,B3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
      = ( times_times_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ).

% minus_mult_commute
thf(fact_6676_minus__mult__commute,axiom,
    ! [A3: int,B3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A3 ) @ B3 )
      = ( times_times_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ).

% minus_mult_commute
thf(fact_6677_minus__divide__right,axiom,
    ! [A3: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
      = ( divide1717551699836669952omplex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).

% minus_divide_right
thf(fact_6678_minus__divide__right,axiom,
    ! [A3: real,B3: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ).

% minus_divide_right
thf(fact_6679_minus__divide__right,axiom,
    ! [A3: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
      = ( divide_divide_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ).

% minus_divide_right
thf(fact_6680_minus__divide__divide,axiom,
    ! [A3: complex,B3: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
      = ( divide1717551699836669952omplex @ A3 @ B3 ) ) ).

% minus_divide_divide
thf(fact_6681_minus__divide__divide,axiom,
    ! [A3: real,B3: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
      = ( divide_divide_real @ A3 @ B3 ) ) ).

% minus_divide_divide
thf(fact_6682_minus__divide__divide,axiom,
    ! [A3: rat,B3: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) )
      = ( divide_divide_rat @ A3 @ B3 ) ) ).

% minus_divide_divide
thf(fact_6683_minus__divide__left,axiom,
    ! [A3: complex,B3: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 ) ) ).

% minus_divide_left
thf(fact_6684_minus__divide__left,axiom,
    ! [A3: real,B3: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).

% minus_divide_left
thf(fact_6685_minus__divide__left,axiom,
    ! [A3: rat,B3: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ).

% minus_divide_left
thf(fact_6686_div__minus__right,axiom,
    ! [A3: int,B3: int] :
      ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).

% div_minus_right
thf(fact_6687_mod__minus__eq,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ B3 ) ) @ B3 )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).

% mod_minus_eq
thf(fact_6688_mod__minus__eq,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ B3 ) ) @ B3 )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).

% mod_minus_eq
thf(fact_6689_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A3: code_integer,B3: code_integer,A5: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A3 @ B3 )
        = ( modulo364778990260209775nteger @ A5 @ B3 ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A5 ) @ B3 ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_6690_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A3: int,B3: int,A5: int] :
      ( ( ( modulo_modulo_int @ A3 @ B3 )
        = ( modulo_modulo_int @ A5 @ B3 ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A5 ) @ B3 ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
thf(fact_6691_mod__minus__right,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( modulo364778990260209775nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).

% mod_minus_right
thf(fact_6692_mod__minus__right,axiom,
    ! [A3: int,B3: int] :
      ( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ B3 ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).

% mod_minus_right
thf(fact_6693_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6694_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_6695_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6696_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_6697_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_6698_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_6699_curryD,axiom,
    ! [F: product_prod_num_num > $o,A3: num,B3: num] :
      ( ( produc932273227547478743_num_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ).

% curryD
thf(fact_6700_curryD,axiom,
    ! [F: produc8398139464844984134T_VEBT > $o,A3: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( produc1960320730580999890VEBT_o @ F @ A3 @ B3 )
     => ( F @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) ) ) ).

% curryD
thf(fact_6701_curryD,axiom,
    ! [F: product_prod_nat_num > $o,A3: nat,B3: num] :
      ( ( produc156083480235303841_num_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ).

% curryD
thf(fact_6702_curryD,axiom,
    ! [F: product_prod_nat_nat > $o,A3: nat,B3: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ).

% curryD
thf(fact_6703_curryD,axiom,
    ! [F: product_prod_int_int > $o,A3: int,B3: int] :
      ( ( produc175634133007206835_int_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).

% curryD
thf(fact_6704_curryE,axiom,
    ! [F: product_prod_num_num > $o,A3: num,B3: num] :
      ( ( produc932273227547478743_num_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_num_num @ A3 @ B3 ) ) ) ).

% curryE
thf(fact_6705_curryE,axiom,
    ! [F: produc8398139464844984134T_VEBT > $o,A3: nat,B3: produc4813437837504472865T_VEBT] :
      ( ( produc1960320730580999890VEBT_o @ F @ A3 @ B3 )
     => ( F @ ( produc1750349459881913976T_VEBT @ A3 @ B3 ) ) ) ).

% curryE
thf(fact_6706_curryE,axiom,
    ! [F: product_prod_nat_num > $o,A3: nat,B3: num] :
      ( ( produc156083480235303841_num_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_nat_num @ A3 @ B3 ) ) ) ).

% curryE
thf(fact_6707_curryE,axiom,
    ! [F: product_prod_nat_nat > $o,A3: nat,B3: nat] :
      ( ( produc1310100445399344235_nat_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_nat_nat @ A3 @ B3 ) ) ) ).

% curryE
thf(fact_6708_curryE,axiom,
    ! [F: product_prod_int_int > $o,A3: int,B3: int] :
      ( ( produc175634133007206835_int_o @ F @ A3 @ B3 )
     => ( F @ ( product_Pair_int_int @ A3 @ B3 ) ) ) ).

% curryE
thf(fact_6709_TBOUND__empty,axiom,
    ! [T2: nat] :
      ( time_TBOUND_nat
      @ ( heap_Time_Heap_nat2
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P281974696781278558it_nat )
      @ T2 ) ).

% TBOUND_empty
thf(fact_6710_TBOUND__empty,axiom,
    ! [T2: nat] :
      ( time_T8353473612707095248on_nat
      @ ( heap_T5286843759275942675on_nat
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat )
      @ T2 ) ).

% TBOUND_empty
thf(fact_6711_TBOUND__empty,axiom,
    ! [T2: nat] :
      ( time_TBOUND_o
      @ ( heap_Time_Heap_o2
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat )
      @ T2 ) ).

% TBOUND_empty
thf(fact_6712_TBOUND__empty,axiom,
    ! [T2: nat] :
      ( time_T5737551269749752165_VEBTi
      @ ( heap_T1489671670754571048_VEBTi
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat )
      @ T2 ) ).

% TBOUND_empty
thf(fact_6713_refines__empty,axiom,
    ! [M: heap_T2636463487746394924on_nat] :
      ( refine7594492741263601813on_nat @ M
      @ ( heap_T5286843759275942675on_nat
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) ) ).

% refines_empty
thf(fact_6714_refines__empty,axiom,
    ! [M: heap_Time_Heap_o] :
      ( refine_Imp_refines_o @ M
      @ ( heap_Time_Heap_o2
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) ) ).

% refines_empty
thf(fact_6715_refines__empty,axiom,
    ! [M: heap_T8145700208782473153_VEBTi] :
      ( refine5565527176597971370_VEBTi @ M
      @ ( heap_T1489671670754571048_VEBTi
        @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) ) ).

% refines_empty
thf(fact_6716_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6717_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6718_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6719_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_6720_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_6721_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_6722_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_6723_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6724_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6725_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_6726_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_6727_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_6728_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_6729_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6730_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6731_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_6732_neg__eq__iff__add__eq__0,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( uminus8244633308260627903l_num1 @ A3 )
        = B3 )
      = ( ( plus_p361126936061061375l_num1 @ A3 @ B3 )
        = zero_z3563351764282998399l_num1 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6733_neg__eq__iff__add__eq__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ( uminus1482373934393186551omplex @ A3 )
        = B3 )
      = ( ( plus_plus_complex @ A3 @ B3 )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6734_neg__eq__iff__add__eq__0,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ( uminus_uminus_uint32 @ A3 )
        = B3 )
      = ( ( plus_plus_uint32 @ A3 @ B3 )
        = zero_zero_uint32 ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6735_neg__eq__iff__add__eq__0,axiom,
    ! [A3: real,B3: real] :
      ( ( ( uminus_uminus_real @ A3 )
        = B3 )
      = ( ( plus_plus_real @ A3 @ B3 )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6736_neg__eq__iff__add__eq__0,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( uminus_uminus_rat @ A3 )
        = B3 )
      = ( ( plus_plus_rat @ A3 @ B3 )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6737_neg__eq__iff__add__eq__0,axiom,
    ! [A3: int,B3: int] :
      ( ( ( uminus_uminus_int @ A3 )
        = B3 )
      = ( ( plus_plus_int @ A3 @ B3 )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_6738_eq__neg__iff__add__eq__0,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( A3
        = ( uminus8244633308260627903l_num1 @ B3 ) )
      = ( ( plus_p361126936061061375l_num1 @ A3 @ B3 )
        = zero_z3563351764282998399l_num1 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6739_eq__neg__iff__add__eq__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( A3
        = ( uminus1482373934393186551omplex @ B3 ) )
      = ( ( plus_plus_complex @ A3 @ B3 )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6740_eq__neg__iff__add__eq__0,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( A3
        = ( uminus_uminus_uint32 @ B3 ) )
      = ( ( plus_plus_uint32 @ A3 @ B3 )
        = zero_zero_uint32 ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6741_eq__neg__iff__add__eq__0,axiom,
    ! [A3: real,B3: real] :
      ( ( A3
        = ( uminus_uminus_real @ B3 ) )
      = ( ( plus_plus_real @ A3 @ B3 )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6742_eq__neg__iff__add__eq__0,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3
        = ( uminus_uminus_rat @ B3 ) )
      = ( ( plus_plus_rat @ A3 @ B3 )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6743_eq__neg__iff__add__eq__0,axiom,
    ! [A3: int,B3: int] :
      ( ( A3
        = ( uminus_uminus_int @ B3 ) )
      = ( ( plus_plus_int @ A3 @ B3 )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_6744_add_Oinverse__unique,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A3 @ B3 )
        = zero_z3563351764282998399l_num1 )
     => ( ( uminus8244633308260627903l_num1 @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6745_add_Oinverse__unique,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ( plus_plus_complex @ A3 @ B3 )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6746_add_Oinverse__unique,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ( plus_plus_uint32 @ A3 @ B3 )
        = zero_zero_uint32 )
     => ( ( uminus_uminus_uint32 @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6747_add_Oinverse__unique,axiom,
    ! [A3: real,B3: real] :
      ( ( ( plus_plus_real @ A3 @ B3 )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6748_add_Oinverse__unique,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A3 @ B3 )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6749_add_Oinverse__unique,axiom,
    ! [A3: int,B3: int] :
      ( ( ( plus_plus_int @ A3 @ B3 )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A3 )
        = B3 ) ) ).

% add.inverse_unique
thf(fact_6750_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( plus_p361126936061061375l_num1 @ ( uminus8244633308260627903l_num1 @ A3 ) @ A3 )
      = zero_z3563351764282998399l_num1 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6751_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ A3 )
      = zero_zero_complex ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6752_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: uint32] :
      ( ( plus_plus_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ A3 )
      = zero_zero_uint32 ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6753_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ A3 )
      = zero_zero_real ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6754_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ A3 )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6755_ab__group__add__class_Oab__left__minus,axiom,
    ! [A3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A3 ) @ A3 )
      = zero_zero_int ) ).

% ab_group_add_class.ab_left_minus
thf(fact_6756_add__eq__0__iff,axiom,
    ! [A3: word_N3645301735248828278l_num1,B3: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ A3 @ B3 )
        = zero_z3563351764282998399l_num1 )
      = ( B3
        = ( uminus8244633308260627903l_num1 @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6757_add__eq__0__iff,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ( plus_plus_complex @ A3 @ B3 )
        = zero_zero_complex )
      = ( B3
        = ( uminus1482373934393186551omplex @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6758_add__eq__0__iff,axiom,
    ! [A3: uint32,B3: uint32] :
      ( ( ( plus_plus_uint32 @ A3 @ B3 )
        = zero_zero_uint32 )
      = ( B3
        = ( uminus_uminus_uint32 @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6759_add__eq__0__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( plus_plus_real @ A3 @ B3 )
        = zero_zero_real )
      = ( B3
        = ( uminus_uminus_real @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6760_add__eq__0__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( plus_plus_rat @ A3 @ B3 )
        = zero_zero_rat )
      = ( B3
        = ( uminus_uminus_rat @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6761_add__eq__0__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ( plus_plus_int @ A3 @ B3 )
        = zero_zero_int )
      = ( B3
        = ( uminus_uminus_int @ A3 ) ) ) ).

% add_eq_0_iff
thf(fact_6762_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_6763_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_6764_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_6765_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_6766_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_6767_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_6768_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_6769_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_6770_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_6771_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_6772_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_6773_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_6774_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_6775_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_6776_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_6777_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_6778_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ N2 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_6779_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ N2 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_6780_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_6781_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_6782_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_6783_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_6784_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_6785_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_6786_numeral__times__minus__swap,axiom,
    ! [W: num,X2: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ W ) @ ( uminus8244633308260627903l_num1 @ X2 ) )
      = ( times_7065122842183080059l_num1 @ X2 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6787_numeral__times__minus__swap,axiom,
    ! [W: num,X2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X2 ) )
      = ( times_times_complex @ X2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6788_numeral__times__minus__swap,axiom,
    ! [W: num,X2: uint32] :
      ( ( times_times_uint32 @ ( numera9087168376688890119uint32 @ W ) @ ( uminus_uminus_uint32 @ X2 ) )
      = ( times_times_uint32 @ X2 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6789_numeral__times__minus__swap,axiom,
    ! [W: num,X2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X2 ) )
      = ( times_times_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6790_numeral__times__minus__swap,axiom,
    ! [W: num,X2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X2 ) )
      = ( times_times_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6791_numeral__times__minus__swap,axiom,
    ! [W: num,X2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X2 ) )
      = ( times_times_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_6792_nonzero__minus__divide__right,axiom,
    ! [B3: complex,A3: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
        = ( divide1717551699836669952omplex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_6793_nonzero__minus__divide__right,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
        = ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_6794_nonzero__minus__divide__right,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
        = ( divide_divide_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_6795_nonzero__minus__divide__divide,axiom,
    ! [B3: complex,A3: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ ( uminus1482373934393186551omplex @ B3 ) )
        = ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_6796_nonzero__minus__divide__divide,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ ( uminus_uminus_real @ B3 ) )
        = ( divide_divide_real @ A3 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_6797_nonzero__minus__divide__divide,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ ( uminus_uminus_rat @ B3 ) )
        = ( divide_divide_rat @ A3 @ B3 ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_6798_square__eq__1__iff,axiom,
    ! [X2: complex] :
      ( ( ( times_times_complex @ X2 @ X2 )
        = one_one_complex )
      = ( ( X2 = one_one_complex )
        | ( X2
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_6799_square__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( times_times_real @ X2 @ X2 )
        = one_one_real )
      = ( ( X2 = one_one_real )
        | ( X2
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_6800_square__eq__1__iff,axiom,
    ! [X2: rat] :
      ( ( ( times_times_rat @ X2 @ X2 )
        = one_one_rat )
      = ( ( X2 = one_one_rat )
        | ( X2
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_6801_square__eq__1__iff,axiom,
    ! [X2: int] :
      ( ( ( times_times_int @ X2 @ X2 )
        = one_one_int )
      = ( ( X2 = one_one_int )
        | ( X2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_6802_dvd__div__neg,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ( ( divide6298287555418463151nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).

% dvd_div_neg
thf(fact_6803_dvd__div__neg,axiom,
    ! [B3: complex,A3: complex] :
      ( ( dvd_dvd_complex @ B3 @ A3 )
     => ( ( divide1717551699836669952omplex @ A3 @ ( uminus1482373934393186551omplex @ B3 ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ) ).

% dvd_div_neg
thf(fact_6804_dvd__div__neg,axiom,
    ! [B3: real,A3: real] :
      ( ( dvd_dvd_real @ B3 @ A3 )
     => ( ( divide_divide_real @ A3 @ ( uminus_uminus_real @ B3 ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ) ).

% dvd_div_neg
thf(fact_6805_dvd__div__neg,axiom,
    ! [B3: rat,A3: rat] :
      ( ( dvd_dvd_rat @ B3 @ A3 )
     => ( ( divide_divide_rat @ A3 @ ( uminus_uminus_rat @ B3 ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).

% dvd_div_neg
thf(fact_6806_dvd__div__neg,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% dvd_div_neg
thf(fact_6807_dvd__neg__div,axiom,
    ! [B3: code_integer,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ B3 @ A3 )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A3 @ B3 ) ) ) ) ).

% dvd_neg_div
thf(fact_6808_dvd__neg__div,axiom,
    ! [B3: complex,A3: complex] :
      ( ( dvd_dvd_complex @ B3 @ A3 )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A3 ) @ B3 )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) ) ) ).

% dvd_neg_div
thf(fact_6809_dvd__neg__div,axiom,
    ! [B3: real,A3: real] :
      ( ( dvd_dvd_real @ B3 @ A3 )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A3 ) @ B3 )
        = ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) ) ) ).

% dvd_neg_div
thf(fact_6810_dvd__neg__div,axiom,
    ! [B3: rat,A3: rat] :
      ( ( dvd_dvd_rat @ B3 @ A3 )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) ) ) ).

% dvd_neg_div
thf(fact_6811_dvd__neg__div,axiom,
    ! [B3: int,A3: int] :
      ( ( dvd_dvd_int @ B3 @ A3 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 )
        = ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) ) ).

% dvd_neg_div
thf(fact_6812_Heap_Osize_I2_J,axiom,
    ! [X2: heap_e7401611519738050253t_unit > option2621746655072343315it_nat] :
      ( ( size_s6287829766004316056on_nat @ ( heap_T5286843759275942675on_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size(2)
thf(fact_6813_Heap_Osize_I2_J,axiom,
    ! [X2: heap_e7401611519738050253t_unit > option7339022715339332451it_nat] :
      ( ( size_s2700093152935483318Heap_o @ ( heap_Time_Heap_o2 @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size(2)
thf(fact_6814_Heap_Osize_I2_J,axiom,
    ! [X2: heap_e7401611519738050253t_unit > option5408194888911472936it_nat] :
      ( ( size_s8425857057747876397_VEBTi @ ( heap_T1489671670754571048_VEBTi @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% Heap.size(2)
thf(fact_6815_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_le_zero
thf(fact_6816_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_6817_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_le_zero
thf(fact_6818_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6819_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6820_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6821_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).

% neg_numeral_less_zero
thf(fact_6822_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_6823_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).

% neg_numeral_less_zero
thf(fact_6824_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6825_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6826_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6827_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_6828_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_6829_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_6830_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_6831_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_6832_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_6833_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_6834_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_6835_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_6836_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_6837_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_6838_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_6839_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_6840_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_6841_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_6842_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_6843_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_6844_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_6845_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_6846_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_6847_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_6848_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_6849_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_6850_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_6851_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_6852_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_6853_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_6854_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_6855_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_6856_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_6857_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_6858_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_6859_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_6860_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_6861_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_6862_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_6863_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_6864_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_6865_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_6866_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_6867_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_6868_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_6869_uminus__numeral__One,axiom,
    ( ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% uminus_numeral_One
thf(fact_6870_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_6871_uminus__numeral__One,axiom,
    ( ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% uminus_numeral_One
thf(fact_6872_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_6873_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_6874_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_6875_mult__1s__ring__1_I2_J,axiom,
    ! [B3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ B3 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) ) )
      = ( uminus8244633308260627903l_num1 @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6876_mult__1s__ring__1_I2_J,axiom,
    ! [B3: complex] :
      ( ( times_times_complex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6877_mult__1s__ring__1_I2_J,axiom,
    ! [B3: uint32] :
      ( ( times_times_uint32 @ B3 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) )
      = ( uminus_uminus_uint32 @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6878_mult__1s__ring__1_I2_J,axiom,
    ! [B3: real] :
      ( ( times_times_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6879_mult__1s__ring__1_I2_J,axiom,
    ! [B3: rat] :
      ( ( times_times_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6880_mult__1s__ring__1_I2_J,axiom,
    ! [B3: int] :
      ( ( times_times_int @ B3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B3 ) ) ).

% mult_1s_ring_1(2)
thf(fact_6881_mult__1s__ring__1_I1_J,axiom,
    ! [B3: word_N3645301735248828278l_num1] :
      ( ( times_7065122842183080059l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ one ) ) @ B3 )
      = ( uminus8244633308260627903l_num1 @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6882_mult__1s__ring__1_I1_J,axiom,
    ! [B3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B3 )
      = ( uminus1482373934393186551omplex @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6883_mult__1s__ring__1_I1_J,axiom,
    ! [B3: uint32] :
      ( ( times_times_uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ one ) ) @ B3 )
      = ( uminus_uminus_uint32 @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6884_mult__1s__ring__1_I1_J,axiom,
    ! [B3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B3 )
      = ( uminus_uminus_real @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6885_mult__1s__ring__1_I1_J,axiom,
    ! [B3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B3 )
      = ( uminus_uminus_rat @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6886_mult__1s__ring__1_I1_J,axiom,
    ! [B3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B3 )
      = ( uminus_uminus_int @ B3 ) ) ).

% mult_1s_ring_1(1)
thf(fact_6887_divide__eq__minus__1__iff,axiom,
    ! [A3: complex,B3: complex] :
      ( ( ( divide1717551699836669952omplex @ A3 @ B3 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B3 != zero_zero_complex )
        & ( A3
          = ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_6888_divide__eq__minus__1__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ( divide_divide_real @ A3 @ B3 )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B3 != zero_zero_real )
        & ( A3
          = ( uminus_uminus_real @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_6889_divide__eq__minus__1__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( divide_divide_rat @ A3 @ B3 )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B3 != zero_zero_rat )
        & ( A3
          = ( uminus_uminus_rat @ B3 ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_6890_eq__minus__divide__eq,axiom,
    ! [A3: complex,B3: complex,C: complex] :
      ( ( A3
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A3 @ C )
            = ( uminus1482373934393186551omplex @ B3 ) ) )
        & ( ( C = zero_zero_complex )
         => ( A3 = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6891_eq__minus__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( A3
        = ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A3 @ C )
            = ( uminus_uminus_real @ B3 ) ) )
        & ( ( C = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6892_eq__minus__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( A3
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A3 @ C )
            = ( uminus_uminus_rat @ B3 ) ) )
        & ( ( C = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_6893_minus__divide__eq__eq,axiom,
    ! [B3: complex,C: complex,A3: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) )
        = A3 )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B3 )
            = ( times_times_complex @ A3 @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A3 = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6894_minus__divide__eq__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) )
        = A3 )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B3 )
            = ( times_times_real @ A3 @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A3 = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6895_minus__divide__eq__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) )
        = A3 )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B3 )
            = ( times_times_rat @ A3 @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A3 = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_6896_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: complex,A3: complex,C: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A3 )
          = ( times_times_complex @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6897_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: real,A3: real,C: real] :
      ( ( B3 != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) )
          = C )
        = ( ( uminus_uminus_real @ A3 )
          = ( times_times_real @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6898_nonzero__neg__divide__eq__eq,axiom,
    ! [B3: rat,A3: rat,C: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) )
          = C )
        = ( ( uminus_uminus_rat @ A3 )
          = ( times_times_rat @ C @ B3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_6899_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: complex,C: complex,A3: complex] :
      ( ( B3 != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ B3 ) ) )
        = ( ( times_times_complex @ C @ B3 )
          = ( uminus1482373934393186551omplex @ A3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6900_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A3 @ B3 ) ) )
        = ( ( times_times_real @ C @ B3 )
          = ( uminus_uminus_real @ A3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6901_nonzero__neg__divide__eq__eq2,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ B3 ) ) )
        = ( ( times_times_rat @ C @ B3 )
          = ( uminus_uminus_rat @ A3 ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_6902_power__minus,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6903_power__minus,axiom,
    ! [A3: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6904_power__minus,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 )
      = ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 ) @ ( power_power_uint32 @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6905_power__minus,axiom,
    ! [A3: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6906_power__minus,axiom,
    ! [A3: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6907_power__minus,axiom,
    ! [A3: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A3 @ N2 ) ) ) ).

% power_minus
thf(fact_6908_power__minus__Bit0,axiom,
    ! [X2: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6909_power__minus__Bit0,axiom,
    ! [X2: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6910_power__minus__Bit0,axiom,
    ! [X2: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_uint32 @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6911_power__minus__Bit0,axiom,
    ! [X2: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6912_power__minus__Bit0,axiom,
    ! [X2: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6913_power__minus__Bit0,axiom,
    ! [X2: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_6914_power__minus__Bit1,axiom,
    ! [X2: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6915_power__minus__Bit1,axiom,
    ! [X2: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6916_power__minus__Bit1,axiom,
    ! [X2: uint32,K: num] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( power_power_uint32 @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6917_power__minus__Bit1,axiom,
    ! [X2: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6918_power__minus__Bit1,axiom,
    ! [X2: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6919_power__minus__Bit1,axiom,
    ! [X2: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_6920_real__add__less__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
      = ( ord_less_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_less_0_iff
thf(fact_6921_real__0__less__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).

% real_0_less_add_iff
thf(fact_6922_real__0__le__add__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y2 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y2 ) ) ).

% real_0_le_add_iff
thf(fact_6923_real__add__le__0__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y2 @ ( uminus_uminus_real @ X2 ) ) ) ).

% real_add_le_0_iff
thf(fact_6924_tanh__real__gt__neg1,axiom,
    ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).

% tanh_real_gt_neg1
thf(fact_6925_less__minus__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_6926_less__minus__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_6927_minus__divide__less__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_6928_minus__divide__less__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_6929_neg__less__minus__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_6930_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_6931_neg__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
        = ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_6932_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
        = ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_6933_pos__less__minus__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_6934_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_6935_pos__minus__divide__less__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
        = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_6936_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_6937_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B3 @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B3
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6938_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B3 @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B3
            = ( 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_6939_divide__eq__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B3 @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B3
            = ( 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_6940_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B3 @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6941_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B3 @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6942_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B3 ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6943_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A3: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6944_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6945_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
          = B3 ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A3 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6946_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6947_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6948_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6949_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A3: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
          = ( uminus1482373934393186551omplex @ B3 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A3 @ Z ) ) @ B3 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A3 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6950_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
          = ( uminus_uminus_real @ B3 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A3 @ Z ) ) @ B3 )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A3 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6951_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
          = ( uminus_uminus_rat @ B3 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A3 @ Z ) ) @ B3 )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A3 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_6952_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A3: complex,B3: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
          = ( uminus1482373934393186551omplex @ B3 ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A3 @ Z ) @ B3 )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A3 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6953_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A3: real,B3: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
          = ( uminus_uminus_real @ B3 ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A3 @ Z ) @ B3 )
          = ( divide_divide_real @ ( minus_minus_real @ A3 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6954_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A3: rat,B3: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
          = ( uminus_uminus_rat @ B3 ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A3 @ Z ) @ B3 )
          = ( divide_divide_rat @ ( minus_minus_rat @ A3 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_6955_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X2: complex,Y2: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y2 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6956_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X2: real,Y2: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6957_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X2: rat,Y2: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y2 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y2 @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_6958_even__minus,axiom,
    ! [A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A3 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_minus
thf(fact_6959_even__minus,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( uminus8244633308260627903l_num1 @ A3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_minus
thf(fact_6960_even__minus,axiom,
    ! [A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( uminus_uminus_uint32 @ A3 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_minus
thf(fact_6961_even__minus,axiom,
    ! [A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A3 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_minus
thf(fact_6962_power2__eq__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus1351360451143612070nteger @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_6963_power2__eq__iff,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( ( power_power_complex @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus1482373934393186551omplex @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_6964_power2__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_real @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_6965_power2__eq__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_rat @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_6966_power2__eq__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_int @ Y2 ) ) ) ) ).

% power2_eq_iff
thf(fact_6967_pred__equals__eq2,axiom,
    ! [R4: set_Pr8218934625190621173um_num,S4: set_Pr8218934625190621173um_num] :
      ( ( ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ R4 ) )
        = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ S4 ) ) )
      = ( R4 = S4 ) ) ).

% pred_equals_eq2
thf(fact_6968_pred__equals__eq2,axiom,
    ! [R4: set_Pr563407847431865468T_VEBT,S4: set_Pr563407847431865468T_VEBT] :
      ( ( ( ^ [X: nat,Y: produc4813437837504472865T_VEBT] : ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) @ R4 ) )
        = ( ^ [X: nat,Y: produc4813437837504472865T_VEBT] : ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) @ S4 ) ) )
      = ( R4 = S4 ) ) ).

% pred_equals_eq2
thf(fact_6969_pred__equals__eq2,axiom,
    ! [R4: set_Pr6200539531224447659at_num,S4: set_Pr6200539531224447659at_num] :
      ( ( ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ R4 ) )
        = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ S4 ) ) )
      = ( R4 = S4 ) ) ).

% pred_equals_eq2
thf(fact_6970_pred__equals__eq2,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R4 ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S4 ) ) )
      = ( R4 = S4 ) ) ).

% pred_equals_eq2
thf(fact_6971_pred__equals__eq2,axiom,
    ! [R4: set_Pr958786334691620121nt_int,S4: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R4 ) )
        = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S4 ) ) )
      = ( R4 = S4 ) ) ).

% pred_equals_eq2
thf(fact_6972_le__minus__divide__eq,axiom,
    ! [A3: real,B3: real,C: real] :
      ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A3 @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6973_le__minus__divide__eq,axiom,
    ! [A3: rat,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A3 @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6974_minus__divide__le__eq,axiom,
    ! [B3: real,C: real,A3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6975_minus__divide__le__eq,axiom,
    ! [B3: rat,C: rat,A3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6976_neg__le__minus__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6977_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6978_neg__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
        = ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6979_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
        = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6980_pos__le__minus__divide__eq,axiom,
    ! [C: real,A3: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A3 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6981_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A3: rat,B3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A3 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A3 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6982_pos__minus__divide__le__eq,axiom,
    ! [C: real,B3: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A3 )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A3 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6983_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B3: rat,A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A3 )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A3 @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6984_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_6985_divide__less__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_6986_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B3 @ ( 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_6987_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B3 @ ( 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_6988_power2__eq__1__iff,axiom,
    ! [A3: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A3 = one_one_Code_integer )
        | ( A3
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6989_power2__eq__1__iff,axiom,
    ! [A3: complex] :
      ( ( ( power_power_complex @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A3 = one_one_complex )
        | ( A3
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6990_power2__eq__1__iff,axiom,
    ! [A3: real] :
      ( ( ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A3 = one_one_real )
        | ( A3
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6991_power2__eq__1__iff,axiom,
    ! [A3: rat] :
      ( ( ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A3 = one_one_rat )
        | ( A3
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6992_power2__eq__1__iff,axiom,
    ! [A3: int] :
      ( ( ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A3 = one_one_int )
        | ( A3
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_6993_uminus__power__if,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 )
          = ( power_8256067586552552935nteger @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6994_uminus__power__if,axiom,
    ! [N2: nat,A3: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 )
          = ( power_power_complex @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6995_uminus__power__if,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 )
          = ( power_power_uint32 @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 )
          = ( uminus_uminus_uint32 @ ( power_power_uint32 @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6996_uminus__power__if,axiom,
    ! [N2: nat,A3: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 )
          = ( power_power_real @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 )
          = ( uminus_uminus_real @ ( power_power_real @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6997_uminus__power__if,axiom,
    ! [N2: nat,A3: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 )
          = ( power_power_rat @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 )
          = ( uminus_uminus_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6998_uminus__power__if,axiom,
    ! [N2: nat,A3: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 )
          = ( power_power_int @ A3 @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 )
          = ( uminus_uminus_int @ ( power_power_int @ A3 @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_6999_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7000_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7001_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7002_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7003_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7004_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_7005_realpow__square__minus__le,axiom,
    ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_7006_ln__add__one__self__le__self2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).

% ln_add_one_self_le_self2
thf(fact_7007_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_7008_divide__le__eq__numeral_I2_J,axiom,
    ! [B3: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B3 @ ( 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 ) @ B3 ) )
            & ( ~ ( 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_7009_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B3 @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B3 @ ( 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_7010_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B3: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B3 @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B3 ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B3 @ ( 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_7011_square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
     => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_7012_square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_7013_square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
     => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_7014_square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
     => ( ( ord_less_eq_int @ X2 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_7015_minus__power__mult__self,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7016_minus__power__mult__self,axiom,
    ! [A3: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A3 ) @ N2 ) )
      = ( power_power_complex @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7017_minus__power__mult__self,axiom,
    ! [A3: uint32,N2: nat] :
      ( ( times_times_uint32 @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 ) @ ( power_power_uint32 @ ( uminus_uminus_uint32 @ A3 ) @ N2 ) )
      = ( power_power_uint32 @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7018_minus__power__mult__self,axiom,
    ! [A3: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 ) )
      = ( power_power_real @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7019_minus__power__mult__self,axiom,
    ! [A3: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 ) )
      = ( power_power_rat @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7020_minus__power__mult__self,axiom,
    ! [A3: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 ) )
      = ( power_power_int @ A3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_7021_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_7022_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_7023_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 )
          = one_one_uint32 ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ N2 )
          = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ) ) ).

% minus_one_power_iff
thf(fact_7024_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_7025_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_7026_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_7027_ln__one__minus__pos__upper__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_7028_vebt__memberi_Omono,axiom,
    ! [X2: produc3881548065746020326Ti_nat] :
      ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
      @ ^ [Vebt_memberi3: produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
          ( produc770043135277712853Heap_o
          @ ^ [T: vEBT_VEBTi,X: nat] :
              ( vEBT_c6104975476656191286Heap_o
              @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                  ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                  @ ( produc3505292621261808240Heap_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                        @ ( if_Heap_Time_Heap_o @ ( X = Mi3 ) @ ( heap_Time_return_o @ $true )
                          @ ( if_Heap_Time_Heap_o @ ( X = Ma3 ) @ ( heap_Time_return_o @ $true )
                            @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_o @ $false )
                              @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X ) @ ( heap_Time_return_o @ $false )
                                @ ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [H: nat] :
                                      ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [L: nat] :
                                          ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList )
                                          @ ^ [Len: nat] :
                                              ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                              @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList @ H )
                                                @ ^ [Th: vEBT_VEBTi] : ( produc5685940877448195828Heap_o @ Vebt_memberi3 @ Th @ L ) )
                                              @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                  @ Info2 )
              @ ^ [A2: $o,B2: $o] :
                  ( heap_Time_return_o
                  @ ( ( ( X = zero_zero_nat )
                     => A2 )
                    & ( ( X != zero_zero_nat )
                     => ( ( ( X = one_one_nat )
                         => B2 )
                        & ( X = one_one_nat ) ) ) ) )
              @ T )
          @ X2 ) ) ).

% vebt_memberi.mono
thf(fact_7029_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_7030_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power_minus1_odd
thf(fact_7031_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% power_minus1_odd
thf(fact_7032_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_real @ one_one_real ) ) ).

% power_minus1_odd
thf(fact_7033_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_7034_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% power_minus1_odd
thf(fact_7035_subrelI,axiom,
    ! [R: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
      ( ! [X3: num,Y3: num] :
          ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X3 @ Y3 ) @ R )
         => ( member7279096912039735102um_num @ ( product_Pair_num_num @ X3 @ Y3 ) @ S ) )
     => ( ord_le880128212290418581um_num @ R @ S ) ) ).

% subrelI
thf(fact_7036_subrelI,axiom,
    ! [R: set_Pr563407847431865468T_VEBT,S: set_Pr563407847431865468T_VEBT] :
      ( ! [X3: nat,Y3: produc4813437837504472865T_VEBT] :
          ( ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X3 @ Y3 ) @ R )
         => ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X3 @ Y3 ) @ S ) )
     => ( ord_le6438908469242860764T_VEBT @ R @ S ) ) ).

% subrelI
thf(fact_7037_subrelI,axiom,
    ! [R: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X3: nat,Y3: num] :
          ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X3 @ Y3 ) @ R )
         => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X3 @ Y3 ) @ S ) )
     => ( ord_le8085105155179020875at_num @ R @ S ) ) ).

% subrelI
thf(fact_7038_subrelI,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% subrelI
thf(fact_7039_subrelI,axiom,
    ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X3: int,Y3: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R @ S ) ) ).

% subrelI
thf(fact_7040_VEBT__internal_Ovebt__memberi_H_Omono,axiom,
    ! [X2: produc3960855945107176009Ti_nat] :
      ( comple6074371103668693207Heap_o @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
      @ ^ [Vebt_memberi4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
          ( produc5872130906356439992Heap_o
          @ ( produc2327743382103342416Heap_o
            @ ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6104975476656191286Heap_o
                @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                    ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
                    @ ^ [Uu: product_unit] :
                        ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                        @ ( produc3505292621261808240Heap_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                              @ ( if_Heap_Time_Heap_o @ ( X = Mi3 ) @ ( heap_Time_return_o @ $true )
                                @ ( if_Heap_Time_Heap_o @ ( X = Ma3 ) @ ( heap_Time_return_o @ $true )
                                  @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                    @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ X ) @ ( heap_Time_return_o @ $false )
                                      @ ( produc1330493526443650053Heap_o
                                        @ ^ [Info3: option4927543243414619207at_nat] :
                                            ( produc5946672270950774087Heap_o
                                            @ ^ [Deg3: nat] :
                                                ( produc5048428016959714504Heap_o
                                                @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                                    ( heap_T7577940988442287570unit_o
                                                    @ ( refine_Imp_assert
                                                      @ ( ( Info2 = Info3 )
                                                        & ( Deg2 = Deg3 ) ) )
                                                    @ ^ [Uv: product_unit] :
                                                        ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                        @ ^ [H: nat] :
                                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [L: nat] :
                                                                ( heap_T7577940988442287570unit_o
                                                                @ ( refine_Imp_assert
                                                                  @ ( ( L
                                                                      = ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                    & ( H
                                                                      = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                @ ^ [Uw: product_unit] :
                                                                    ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                    @ ^ [Len: nat] :
                                                                        ( heap_T7577940988442287570unit_o
                                                                        @ ( refine_Imp_assert
                                                                          @ ( Len
                                                                            = ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                        @ ^ [Ux: product_unit] :
                                                                            ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                                            @ ( heap_T7577940988442287570unit_o
                                                                              @ ( refine_Imp_assert
                                                                                @ ( ( H
                                                                                    = ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                  & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                                                                              @ ^ [Uy: product_unit] :
                                                                                  ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                  @ ^ [Th: vEBT_VEBTi] : ( produc2663629013181010545Heap_o @ ( produc8381543706267210711Heap_o @ Vebt_memberi4 ) @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Th @ L ) ) )
                                                                            @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                        @ ( vEBT_c634343235235684882T_VEBT
                                          @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                          @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                                          @ T ) ) ) ) ) ) ) )
                        @ Info2 ) )
                @ ^ [A2: $o,B2: $o] :
                    ( heap_Time_return_o
                    @ ( ( ( X = zero_zero_nat )
                       => A2 )
                      & ( ( X != zero_zero_nat )
                       => ( ( ( X = one_one_nat )
                           => B2 )
                          & ( X = one_one_nat ) ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_memberi'.mono
thf(fact_7041_of__int__code__if,axiom,
    ( ring_17408606157368542149l_num1
    = ( ^ [K3: int] :
          ( if_wor5778924947035936048l_num1 @ ( K3 = zero_zero_int ) @ zero_z3563351764282998399l_num1
          @ ( if_wor5778924947035936048l_num1 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus8244633308260627903l_num1 @ ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_wor5778924947035936048l_num1
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p361126936061061375l_num1 @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( ring_17408606157368542149l_num1 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_on7727431528512463931l_num1 ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7042_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7043_of__int__code__if,axiom,
    ( ring_1_of_int_uint32
    = ( ^ [K3: int] :
          ( if_uint32 @ ( K3 = zero_zero_int ) @ zero_zero_uint32
          @ ( if_uint32 @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_uint32 @ ( ring_1_of_int_uint32 @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_uint32
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_uint32 @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( ring_1_of_int_uint32 @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_uint32 ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7044_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7045_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7046_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_7047_vebt__predi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > $o] :
      ( ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
        @ ^ [Vebt_predi4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] : ( P @ ( produc1489253303066280154on_nat @ Vebt_predi4 ) ) )
     => ( ( P
          @ ^ [Vebt_predi4: vEBT_VEBTi,T: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F5: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBTi,A2: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                        @ ^ [Mima: product_prod_nat_nat] :
                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ A2 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                              @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                @ ^ [L: nat] :
                                    ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [H: nat] :
                                        ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                        @ ^ [Aktnode: vEBT_VEBTi] :
                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                            @ ^ [Minlow: option_nat] :
                                                ( if_Hea5867803462524415986on_nat
                                                @ ( ( Minlow != none_nat )
                                                  & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( F5 @ Aktnode @ L )
                                                  @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                @ ( heap_T3669509953089699273on_nat @ ( F5 @ Summary2 @ H )
                                                  @ ^ [Predsum: option_nat] :
                                                      ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ A2 ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                      @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                        @ ^ [Nextnode: vEBT_VEBTi] :
                                                            ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                            @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info2 )
                    @ ^ [B2: $o,C3: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( if_Hea5867803462524415986on_nat @ C3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( A2 = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
                    @ X8 ) ) )
         => ( P @ vEBT_vebt_predi ) ) ) ) ).

% vebt_predi.fixp_induct
thf(fact_7048_vebt__succi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > heap_T2636463487746394924on_nat ) > $o] :
      ( ( comple6931689918642796574on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat )
        @ ^ [Vebt_succi4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] : ( P @ ( produc1489253303066280154on_nat @ Vebt_succi4 ) ) )
     => ( ( P
          @ ^ [Vebt_succi4: vEBT_VEBTi,T: nat] :
              ( heap_T5286843759275942675on_nat
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P1551326421579882414it_nat ) )
       => ( ! [F5: vEBT_VEBTi > nat > heap_T2636463487746394924on_nat] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBTi,A2: nat] :
                    ( vEBT_c6250501799366334488on_nat
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                        @ ^ [Mima: product_prod_nat_nat] :
                            ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                            @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ A2 @ ( product_fst_nat_nat @ Mima ) ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) )
                              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( product_snd_nat_nat @ Mima ) @ A2 ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [L: nat] :
                                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [H: nat] :
                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Aktnode )
                                              @ ^ [Maxlow: option_nat] :
                                                  ( if_Hea5867803462524415986on_nat
                                                  @ ( ( Maxlow != none_nat )
                                                    & ( vEBT_VEBT_less @ ( some_nat @ L ) @ Maxlow ) )
                                                  @ ( heap_T3669509953089699273on_nat @ ( F5 @ Aktnode @ L )
                                                    @ ^ [Succy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Succy ) ) )
                                                  @ ( heap_T3669509953089699273on_nat @ ( F5 @ Summary2 @ H )
                                                    @ ^ [Succsum: option_nat] :
                                                        ( if_Hea5867803462524415986on_nat @ ( Succsum = none_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                                        @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Succsum ) )
                                                          @ ^ [Nextnode: vEBT_VEBTi] :
                                                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Nextnode )
                                                              @ ^ [Minnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Succsum ) @ Minnext ) ) ) ) ) ) ) ) ) ) ) ) ) )
                        @ Info2 )
                    @ ^ [B2: $o,C3: $o] : ( if_Hea5867803462524415986on_nat @ ( A2 = zero_zero_nat ) @ ( if_Hea5867803462524415986on_nat @ C3 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                    @ X8 ) ) )
         => ( P @ vEBT_vebt_succi ) ) ) ) ).

% vebt_succi.fixp_induct
thf(fact_7049_compl__le__compl__iff,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( uminus5710092332889474511et_nat @ Y2 ) )
      = ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ).

% compl_le_compl_iff
thf(fact_7050_ln__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X2 )
          = ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7051_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7052_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_7053_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7054_word__of__int__neg__numeral,axiom,
    ! [Bin: num] :
      ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ Bin ) ) )
      = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ Bin ) ) ) ).

% word_of_int_neg_numeral
thf(fact_7055_word__of__int__neg__1,axiom,
    ( ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% word_of_int_neg_1
thf(fact_7056_int__div__minus__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( ( divide_divide_int @ A3 @ B3 )
          = ( uminus_uminus_int @ A3 ) )
        = ( B3
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_div_minus_is_minus1
thf(fact_7057_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7058_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7059_ceiling__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A3 ) @ ( numeral_numeral_real @ B3 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A3 ) ) @ ( numeral_numeral_int @ B3 ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7060_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A3 ) @ ( numeral_numeral_real @ B3 ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A3 ) @ ( numeral_numeral_int @ B3 ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7061_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_7062_uminus__integer__code_I1_J,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% uminus_integer_code(1)
thf(fact_7063_word__neg__numeral__alt,axiom,
    ! [B3: num] :
      ( ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ B3 ) )
      = ( ring_17408606157368542149l_num1 @ ( uminus_uminus_int @ ( numeral_numeral_int @ B3 ) ) ) ) ).

% word_neg_numeral_alt
thf(fact_7064_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_7065_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_7066_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_7067_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_7068_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_7069_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_7070_max__word__not__0,axiom,
    ( ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 )
   != zero_z3563351764282998399l_num1 ) ).

% max_word_not_0
thf(fact_7071_minus__integer__code_I2_J,axiom,
    ! [L2: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
      = ( uminus1351360451143612070nteger @ L2 ) ) ).

% minus_integer_code(2)
thf(fact_7072_int__cases4,axiom,
    ! [M: int] :
      ( ! [N4: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N4 ) )
     => ~ ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).

% int_cases4
thf(fact_7073_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_7074_nonpos__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ~ ! [N4: nat] :
            ( K
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).

% nonpos_int_cases
thf(fact_7075_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_7076_max__word__wrap,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( ( plus_p361126936061061375l_num1 @ X2 @ one_on7727431528512463931l_num1 )
        = zero_z3563351764282998399l_num1 )
     => ( X2
        = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ) ).

% max_word_wrap
thf(fact_7077_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B3: int] :
      ( ( ( ( modulo_modulo_int @ A3 @ B3 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A3 @ B3 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A3 ) @ B3 )
          = ( minus_minus_int @ B3 @ ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_7078_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B3: int] :
      ( ( ( ( modulo_modulo_int @ A3 @ B3 )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ B3 ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A3 @ B3 )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A3 @ ( uminus_uminus_int @ B3 ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A3 @ B3 ) @ B3 ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_7079_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N4: nat] :
            ( ( K
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
       => ~ ! [N4: nat] :
              ( ( K
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).

% int_cases3
thf(fact_7080_not__zle__0__negative,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).

% not_zle_0_negative
thf(fact_7081_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7082_negD,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ zero_zero_int )
     => ? [N4: nat] :
          ( X2
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).

% negD
thf(fact_7083_verit__less__mono__div__int2,axiom,
    ! [A4: int,B6: int,N2: int] :
      ( ( ord_less_eq_int @ A4 @ B6 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B6 @ N2 ) @ ( divide_divide_int @ A4 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_7084_div__eq__minus1,axiom,
    ! [B3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_7085_ceiling__divide__eq__div,axiom,
    ! [A3: int,B3: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A3 ) @ ( ring_1_of_int_real @ B3 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7086_ceiling__divide__eq__div,axiom,
    ! [A3: int,B3: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A3 ) @ ( ring_1_of_int_rat @ B3 ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7087_neg__int__cases,axiom,
    ! [K: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ~ ! [N4: nat] :
            ( ( K
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).

% neg_int_cases
thf(fact_7088_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_7089_zmod__minus1,axiom,
    ! [B3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
        = ( minus_minus_int @ B3 @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_7090_zdiv__zminus2__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A3 @ B3 )
            = zero_zero_int )
         => ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) )
        & ( ( ( modulo_modulo_int @ A3 @ B3 )
           != zero_zero_int )
         => ( ( divide_divide_int @ A3 @ ( uminus_uminus_int @ B3 ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_7091_zdiv__zminus1__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A3 @ B3 )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 )
            = ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) ) )
        & ( ( ( modulo_modulo_int @ A3 @ B3 )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A3 ) @ B3 )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A3 @ B3 ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_7092_zminus1__lemma,axiom,
    ! [A3: int,B3: int,Q2: int,R: int] :
      ( ( eucl_rel_int @ A3 @ B3 @ ( product_Pair_int_int @ Q2 @ R ) )
     => ( ( B3 != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A3 ) @ B3 @ ( 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 @ B3 @ R ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_7093_minus__1__div__exp__eq__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_7094_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_7095_compl__mono,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ Y2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ ( uminus5710092332889474511et_nat @ X2 ) ) ) ).

% compl_mono
thf(fact_7096_compl__le__swap1,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y2 @ ( uminus5710092332889474511et_nat @ X2 ) )
     => ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y2 ) ) ) ).

% compl_le_swap1
thf(fact_7097_compl__le__swap2,axiom,
    ! [Y2: set_nat,X2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y2 ) @ X2 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y2 ) ) ).

% compl_le_swap2
thf(fact_7098_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_7099_m1mod2k,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ).

% m1mod2k
thf(fact_7100_sb__dec__lem_H,axiom,
    ! [K: nat,A3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) @ A3 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A3 ) ) ) ).

% sb_dec_lem'
thf(fact_7101_m1mod22k,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ one_one_int ) ) ).

% m1mod22k
thf(fact_7102_sb__inc__lem_H,axiom,
    ! [A3: int,K: nat] :
      ( ( ord_less_int @ A3 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ K ) ) ) ) ) ).

% sb_inc_lem'
thf(fact_7103_sb__dec__lem,axiom,
    ! [K: nat,A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A3 ) )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ ( plus_plus_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) @ A3 ) ) ) ).

% sb_dec_lem
thf(fact_7104_diff__shunt__var,axiom,
    ! [X2: assn,Y2: assn] :
      ( ( ( minus_minus_assn @ X2 @ Y2 )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_7105_diff__shunt__var,axiom,
    ! [X2: set_real,Y2: set_real] :
      ( ( ( minus_minus_set_real @ X2 @ Y2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_7106_diff__shunt__var,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( ( minus_minus_set_int @ X2 @ Y2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_7107_diff__shunt__var,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( ( minus_minus_set_nat @ X2 @ Y2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_7108_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7109_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7110_minus__one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_7111_one__mod__minus__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_7112_VEBT__internal_Ovebt__memberi_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o ) > $o] :
      ( ( comple2654586775044187945Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o )
        @ ^ [Vebt_memberi4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] : ( P @ ( produc2663629013181010545Heap_o @ ( produc8381543706267210711Heap_o @ Vebt_memberi4 ) ) ) )
     => ( ( P
          @ ^ [Vebt_memberi4: vEBT_VEBT,T: vEBT_VEBTi,Ti3: nat] :
              ( heap_Time_Heap_o2
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) )
       => ( ! [F5: vEBT_VEBT > vEBT_VEBTi > nat > heap_Time_Heap_o] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBT,A2: vEBT_VEBTi,B2: nat] :
                    ( vEBT_c6104975476656191286Heap_o
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( heap_T7577940988442287570unit_o @ ( refine_Imp_assert @ ( vEBT_is_Node @ X8 ) )
                        @ ^ [Uu: product_unit] :
                            ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                            @ ( produc3505292621261808240Heap_o
                              @ ^ [Mi3: nat,Ma3: nat] :
                                  ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                                  @ ( if_Heap_Time_Heap_o @ ( B2 = Mi3 ) @ ( heap_Time_return_o @ $true )
                                    @ ( if_Heap_Time_Heap_o @ ( B2 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                      @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ B2 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                        @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ B2 ) @ ( heap_Time_return_o @ $false )
                                          @ ( produc1330493526443650053Heap_o
                                            @ ^ [Info3: option4927543243414619207at_nat] :
                                                ( produc5946672270950774087Heap_o
                                                @ ^ [Deg3: nat] :
                                                    ( produc5048428016959714504Heap_o
                                                    @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                                        ( heap_T7577940988442287570unit_o
                                                        @ ( refine_Imp_assert
                                                          @ ( ( Info2 = Info3 )
                                                            & ( Deg2 = Deg3 ) ) )
                                                        @ ^ [Uv: product_unit] :
                                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [H: nat] :
                                                                ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                @ ^ [L: nat] :
                                                                    ( heap_T7577940988442287570unit_o
                                                                    @ ( refine_Imp_assert
                                                                      @ ( ( L
                                                                          = ( vEBT_VEBT_low @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                        & ( H
                                                                          = ( vEBT_VEBT_high @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                                                    @ ^ [Uw: product_unit] :
                                                                        ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                        @ ^ [Len: nat] :
                                                                            ( heap_T7577940988442287570unit_o
                                                                            @ ( refine_Imp_assert
                                                                              @ ( Len
                                                                                = ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                            @ ^ [Ux: product_unit] :
                                                                                ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                                                @ ( heap_T7577940988442287570unit_o
                                                                                  @ ( refine_Imp_assert
                                                                                    @ ( ( H
                                                                                        = ( vEBT_VEBT_high @ B2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                                                                      & ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
                                                                                  @ ^ [Uy: product_unit] :
                                                                                      ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                      @ ^ [Th: vEBT_VEBTi] : ( F5 @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Th @ L ) ) )
                                                                                @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) )
                                            @ ( vEBT_c634343235235684882T_VEBT
                                              @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                              @ ^ [C3: $o,D3: $o] : undefi7074909574607331924T_VEBT
                                              @ X8 ) ) ) ) ) ) ) )
                            @ Info2 ) )
                    @ ^ [C3: $o,D3: $o] :
                        ( heap_Time_return_o
                        @ ( ( ( B2 = zero_zero_nat )
                           => C3 )
                          & ( ( B2 != zero_zero_nat )
                           => ( ( ( B2 = one_one_nat )
                               => D3 )
                              & ( B2 = one_one_nat ) ) ) ) )
                    @ A2 ) ) )
         => ( P @ vEBT_V854960066525838166emberi ) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
thf(fact_7113_one__div__minus__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_7114_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7115_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7116_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7117_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_7118_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_7119_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_7120_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_7121_minus__one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_7122_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_7123_bot__option__def,axiom,
    bot_bot_option_num = none_num ).

% bot_option_def
thf(fact_7124_bot__option__def,axiom,
    bot_bot_option_nat = none_nat ).

% bot_option_def
thf(fact_7125_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_7126_bot__empty__eq2,axiom,
    ( bot_bot_num_num_o
    = ( ^ [X: num,Y: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X @ Y ) @ bot_bo9056780473022590049um_num ) ) ) ).

% bot_empty_eq2
thf(fact_7127_bot__empty__eq2,axiom,
    ( bot_bo7529698899530922655VEBT_o
    = ( ^ [X: nat,Y: produc4813437837504472865T_VEBT] : ( member306291179834725981T_VEBT @ ( produc1750349459881913976T_VEBT @ X @ Y ) @ bot_bo9115540109607619856T_VEBT ) ) ) ).

% bot_empty_eq2
thf(fact_7128_bot__empty__eq2,axiom,
    ( bot_bot_nat_num_o
    = ( ^ [X: nat,Y: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X @ Y ) @ bot_bo7038385379056416535at_num ) ) ) ).

% bot_empty_eq2
thf(fact_7129_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_7130_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_7131_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L: int,R6: int] : ( if_int @ ( R6 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R6 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_7132_vebt__memberi_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > heap_Time_Heap_o ) > $o] :
      ( ( comple6491863954676465222Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o )
        @ ^ [Vebt_memberi3: produc3881548065746020326Ti_nat > heap_Time_Heap_o] : ( P @ ( produc5685940877448195828Heap_o @ Vebt_memberi3 ) ) )
     => ( ( P
          @ ^ [Vebt_memberi3: vEBT_VEBTi,T: nat] :
              ( heap_Time_Heap_o2
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7668321371905463026it_nat ) )
       => ( ! [F5: vEBT_VEBTi > nat > heap_Time_Heap_o] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBTi,A2: nat] :
                    ( vEBT_c6104975476656191286Heap_o
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( case_o1442776274061689234at_nat @ ( heap_Time_return_o @ $false )
                        @ ( produc3505292621261808240Heap_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( if_Heap_Time_Heap_o @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_Time_return_o @ $false )
                              @ ( if_Heap_Time_Heap_o @ ( A2 = Mi3 ) @ ( heap_Time_return_o @ $true )
                                @ ( if_Heap_Time_Heap_o @ ( A2 = Ma3 ) @ ( heap_Time_return_o @ $true )
                                  @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ A2 @ Mi3 ) @ ( heap_Time_return_o @ $false )
                                    @ ( if_Heap_Time_Heap_o @ ( ord_less_nat @ Ma3 @ A2 ) @ ( heap_Time_return_o @ $false )
                                      @ ( heap_Time_bind_nat_o @ ( vEBT_VEBT_highi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_Time_bind_nat_o @ ( vEBT_VEBT_lowi @ A2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L: nat] :
                                                ( heap_Time_bind_nat_o @ ( array_len_VEBT_VEBTi @ TreeList )
                                                @ ^ [Len: nat] :
                                                    ( if_Heap_Time_Heap_o @ ( ord_less_nat @ H @ Len )
                                                    @ ( heap_T3040810144269856602EBTi_o @ ( array_nth_VEBT_VEBTi @ TreeList @ H )
                                                      @ ^ [Th: vEBT_VEBTi] : ( F5 @ Th @ L ) )
                                                    @ ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ) ) )
                        @ Info2 )
                    @ ^ [B2: $o,C3: $o] :
                        ( heap_Time_return_o
                        @ ( ( ( A2 = zero_zero_nat )
                           => B2 )
                          & ( ( A2 != zero_zero_nat )
                           => ( ( ( A2 = one_one_nat )
                               => C3 )
                              & ( A2 = one_one_nat ) ) ) ) )
                    @ X8 ) ) )
         => ( P @ vEBT_vebt_memberi ) ) ) ) ).

% vebt_memberi.fixp_induct
thf(fact_7133_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7134_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_7135_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu93272222329896523l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7136_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7137_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_7138_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_7139_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_7140_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_7141_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ one_one_uint32 )
    = one_one_uint32 ) ).

% dbl_dec_simps(3)
thf(fact_7142_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_7143_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_7144_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_7145_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu93272222329896523l_num1 @ zero_z3563351764282998399l_num1 )
    = ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) ) ).

% dbl_dec_simps(2)
thf(fact_7146_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_7147_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu965353292909893953uint32 @ zero_zero_uint32 )
    = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% dbl_dec_simps(2)
thf(fact_7148_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_7149_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_7150_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_7151_pi__neq__zero,axiom,
    pi != zero_zero_real ).

% pi_neq_zero
thf(fact_7152_zero__integer__def,axiom,
    ( zero_z3403309356797280102nteger
    = ( code_integer_of_int @ zero_zero_int ) ) ).

% zero_integer_def
thf(fact_7153_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( ord_less_int @ Xa @ X2 ) ) ).

% less_integer.abs_eq
thf(fact_7154_one__integer__def,axiom,
    ( one_one_Code_integer
    = ( code_integer_of_int @ one_one_int ) ) ).

% one_integer_def
thf(fact_7155_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7156_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7157_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7158_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( ord_less_eq_int @ Xa @ X2 ) ) ).

% less_eq_integer.abs_eq
thf(fact_7159_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: int] :
      ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X2 ) ) ) ).

% divide_integer.abs_eq
thf(fact_7160_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7161_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7162_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_7163_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7164_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_7165_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_7166_dbl__dec__def,axiom,
    ( neg_nu965353292909893953uint32
    = ( ^ [X: uint32] : ( minus_minus_uint32 @ ( plus_plus_uint32 @ X @ X ) @ one_one_uint32 ) ) ) ).

% dbl_dec_def
thf(fact_7167_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_7168_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_7169_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_7170_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_7171_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_7172_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_7173_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_7174_heap_Omono__body__fixp,axiom,
    ! [F6: ( produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat ) > produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] :
      ( ! [X3: produc3881548065746020326Ti_nat] :
          ( comple4655144769394346904on_nat @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_T2636463487746394924on_nat] : ( F6 @ F4 @ X3 ) )
     => ( ( comple6805837186910174120on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ F6 )
        = ( F6 @ ( comple6805837186910174120on_nat @ ( partia2080987842261039902Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia8535946980722491222Ti_nat @ heap_T7875578835903804844on_nat ) @ F6 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_7175_heap_Omono__body__fixp,axiom,
    ! [F6: ( produc3881548065746020326Ti_nat > heap_Time_Heap_o ) > produc3881548065746020326Ti_nat > heap_Time_Heap_o] :
      ( ! [X3: produc3881548065746020326Ti_nat] :
          ( comple4217288648910406772Heap_o @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3881548065746020326Ti_nat > heap_Time_Heap_o] : ( F6 @ F4 @ X3 ) )
     => ( ( comple2405882057716616508Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ F6 )
        = ( F6 @ ( comple2405882057716616508Heap_o @ ( partia5551857090987368152Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia2925774515620677392Ti_nat @ heap_Time_Heap_ord_o ) @ F6 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_7176_heap_Omono__body__fixp,axiom,
    ! [F6: ( produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi ) > produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
      ( ! [X3: produc3960855945107176009Ti_nat] :
          ( comple5606513277678308283_VEBTi @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ heap_T7173139186834293313_VEBTi
          @ ^ [F4: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] : ( F6 @ F4 @ X3 ) )
     => ( ( comple7072962176332223770_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ F6 )
        = ( F6 @ ( comple7072962176332223770_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ F6 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_7177_heap_Omono__body__fixp,axiom,
    ! [F6: ( produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat ) > produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
      ( ! [X3: produc3960855945107176009Ti_nat] :
          ( comple6977564771798581627on_nat @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ heap_T7875578835903804844on_nat
          @ ^ [F4: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] : ( F6 @ F4 @ X3 ) )
     => ( ( comple8068445680736955397on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ F6 )
        = ( F6 @ ( comple8068445680736955397on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat ) @ F6 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_7178_heap_Omono__body__fixp,axiom,
    ! [F6: ( produc3960855945107176009Ti_nat > heap_Time_Heap_o ) > produc3960855945107176009Ti_nat > heap_Time_Heap_o] :
      ( ! [X3: produc3960855945107176009Ti_nat] :
          ( comple6074371103668693207Heap_o @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ heap_Time_Heap_ord_o
          @ ^ [F4: produc3960855945107176009Ti_nat > heap_Time_Heap_o] : ( F6 @ F4 @ X3 ) )
     => ( ( comple3202505432650402847Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ F6 )
        = ( F6 @ ( comple3202505432650402847Heap_o @ ( partia6726927458685305659Ti_nat @ heap_Time_Heap_lub_o ) @ ( partia3290229181235258227Ti_nat @ heap_Time_Heap_ord_o ) @ F6 ) ) ) ) ).

% heap.mono_body_fixp
thf(fact_7179_sin__cos__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% sin_cos_npi
thf(fact_7180_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_7181_signed__take__bit__rec,axiom,
    ( bit_ri1375673916561920181l_num1
    = ( ^ [N: nat,A2: word_N3645301735248828278l_num1] : ( if_wor5778924947035936048l_num1 @ ( N = zero_zero_nat ) @ ( uminus8244633308260627903l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) @ ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide1791077408188789448l_num1 @ A2 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7182_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A2: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7183_signed__take__bit__rec,axiom,
    ( bit_ri6224792872505173163uint32
    = ( ^ [N: nat,A2: uint32] : ( if_uint32 @ ( N = zero_zero_nat ) @ ( uminus_uminus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) @ ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_uint32 @ A2 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7184_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A2: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7185_dbl__simps_I4_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ ( uminus8244633308260627903l_num1 @ one_on7727431528512463931l_num1 ) )
    = ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7186_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7187_dbl__simps_I4_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
    = ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7188_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_7189_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_7190_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_7191_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_7192_abs__abs,axiom,
    ! [A3: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A3 ) )
      = ( abs_abs_real @ A3 ) ) ).

% abs_abs
thf(fact_7193_abs__abs,axiom,
    ! [A3: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A3 ) )
      = ( abs_abs_int @ A3 ) ) ).

% abs_abs
thf(fact_7194_abs__abs,axiom,
    ! [A3: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A3 ) )
      = ( abs_abs_Code_integer @ A3 ) ) ).

% abs_abs
thf(fact_7195_abs__abs,axiom,
    ! [A3: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A3 ) )
      = ( abs_abs_rat @ A3 ) ) ).

% abs_abs
thf(fact_7196_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_7197_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_7198_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_7199_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_7200_abs__0__eq,axiom,
    ! [A3: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A3 ) )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_7201_abs__0__eq,axiom,
    ! [A3: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A3 ) )
      = ( A3 = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_7202_abs__0__eq,axiom,
    ! [A3: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A3 ) )
      = ( A3 = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_7203_abs__0__eq,axiom,
    ! [A3: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A3 ) )
      = ( A3 = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_7204_abs__eq__0,axiom,
    ! [A3: code_integer] :
      ( ( ( abs_abs_Code_integer @ A3 )
        = zero_z3403309356797280102nteger )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_7205_abs__eq__0,axiom,
    ! [A3: real] :
      ( ( ( abs_abs_real @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_7206_abs__eq__0,axiom,
    ! [A3: rat] :
      ( ( ( abs_abs_rat @ A3 )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_7207_abs__eq__0,axiom,
    ! [A3: int] :
      ( ( ( abs_abs_int @ A3 )
        = zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_7208_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_7209_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_7210_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_7211_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_7212_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_numeral
thf(fact_7213_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_numeral
thf(fact_7214_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_numeral
thf(fact_7215_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_numeral
thf(fact_7216_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_7217_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_7218_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_7219_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_7220_abs__mult__self__eq,axiom,
    ! [A3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ A3 ) )
      = ( times_3573771949741848930nteger @ A3 @ A3 ) ) ).

% abs_mult_self_eq
thf(fact_7221_abs__mult__self__eq,axiom,
    ! [A3: real] :
      ( ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ A3 ) )
      = ( times_times_real @ A3 @ A3 ) ) ).

% abs_mult_self_eq
thf(fact_7222_abs__mult__self__eq,axiom,
    ! [A3: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ A3 ) )
      = ( times_times_rat @ A3 @ A3 ) ) ).

% abs_mult_self_eq
thf(fact_7223_abs__mult__self__eq,axiom,
    ! [A3: int] :
      ( ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ A3 ) )
      = ( times_times_int @ A3 @ A3 ) ) ).

% abs_mult_self_eq
thf(fact_7224_abs__divide,axiom,
    ! [A3: real,B3: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A3 @ B3 ) )
      = ( divide_divide_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_divide
thf(fact_7225_abs__divide,axiom,
    ! [A3: rat,B3: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A3 @ B3 ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_divide
thf(fact_7226_abs__minus,axiom,
    ! [A3: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A3 ) )
      = ( abs_abs_Code_integer @ A3 ) ) ).

% abs_minus
thf(fact_7227_abs__minus,axiom,
    ! [A3: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A3 ) )
      = ( abs_abs_complex @ A3 ) ) ).

% abs_minus
thf(fact_7228_abs__minus,axiom,
    ! [A3: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A3 ) )
      = ( abs_abs_real @ A3 ) ) ).

% abs_minus
thf(fact_7229_abs__minus,axiom,
    ! [A3: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A3 ) )
      = ( abs_abs_rat @ A3 ) ) ).

% abs_minus
thf(fact_7230_abs__minus,axiom,
    ! [A3: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A3 ) )
      = ( abs_abs_int @ A3 ) ) ).

% abs_minus
thf(fact_7231_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri1375673916561920181l_num1 @ N2 @ zero_z3563351764282998399l_num1 )
      = zero_z3563351764282998399l_num1 ) ).

% signed_take_bit_of_0
thf(fact_7232_signed__take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_7233_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_7234_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_7235_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_7236_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_7237_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_7238_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_7239_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_7240_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_7241_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_7242_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_7243_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_7244_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_7245_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_7246_dbl__simps_I2_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ zero_z3563351764282998399l_num1 )
    = zero_z3563351764282998399l_num1 ) ).

% dbl_simps(2)
thf(fact_7247_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_7248_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_7249_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_7250_abs__of__nonneg,axiom,
    ! [A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( abs_abs_Code_integer @ A3 )
        = A3 ) ) ).

% abs_of_nonneg
thf(fact_7251_abs__of__nonneg,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( abs_abs_real @ A3 )
        = A3 ) ) ).

% abs_of_nonneg
thf(fact_7252_abs__of__nonneg,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
     => ( ( abs_abs_rat @ A3 )
        = A3 ) ) ).

% abs_of_nonneg
thf(fact_7253_abs__of__nonneg,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( abs_abs_int @ A3 )
        = A3 ) ) ).

% abs_of_nonneg
thf(fact_7254_abs__le__self__iff,axiom,
    ! [A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ A3 )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 ) ) ).

% abs_le_self_iff
thf(fact_7255_abs__le__self__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ A3 )
      = ( ord_less_eq_real @ zero_zero_real @ A3 ) ) ).

% abs_le_self_iff
thf(fact_7256_abs__le__self__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ A3 )
      = ( ord_less_eq_rat @ zero_zero_rat @ A3 ) ) ).

% abs_le_self_iff
thf(fact_7257_abs__le__self__iff,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ A3 )
      = ( ord_less_eq_int @ zero_zero_int @ A3 ) ) ).

% abs_le_self_iff
thf(fact_7258_abs__le__zero__iff,axiom,
    ! [A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ zero_z3403309356797280102nteger )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_7259_abs__le__zero__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% abs_le_zero_iff
thf(fact_7260_abs__le__zero__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_7261_abs__le__zero__iff,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% abs_le_zero_iff
thf(fact_7262_zero__less__abs__iff,axiom,
    ! [A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A3 ) )
      = ( A3 != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_7263_zero__less__abs__iff,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A3 ) )
      = ( A3 != zero_zero_real ) ) ).

% zero_less_abs_iff
thf(fact_7264_zero__less__abs__iff,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A3 ) )
      = ( A3 != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_7265_zero__less__abs__iff,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A3 ) )
      = ( A3 != zero_zero_int ) ) ).

% zero_less_abs_iff
thf(fact_7266_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_neg_numeral
thf(fact_7267_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_neg_numeral
thf(fact_7268_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_neg_numeral
thf(fact_7269_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_neg_numeral
thf(fact_7270_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_7271_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_7272_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_7273_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_7274_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_7275_abs__power__minus,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A3 ) @ N2 ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% abs_power_minus
thf(fact_7276_abs__power__minus,axiom,
    ! [A3: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A3 ) @ N2 ) )
      = ( abs_abs_real @ ( power_power_real @ A3 @ N2 ) ) ) ).

% abs_power_minus
thf(fact_7277_abs__power__minus,axiom,
    ! [A3: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A3 ) @ N2 ) )
      = ( abs_abs_rat @ ( power_power_rat @ A3 @ N2 ) ) ) ).

% abs_power_minus
thf(fact_7278_abs__power__minus,axiom,
    ! [A3: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A3 ) @ N2 ) )
      = ( abs_abs_int @ ( power_power_int @ A3 @ N2 ) ) ) ).

% abs_power_minus
thf(fact_7279_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6224792872505173163uint32 @ ( suc @ N2 ) @ one_one_uint32 )
      = one_one_uint32 ) ).

% signed_take_bit_Suc_1
thf(fact_7280_signed__take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_Suc_1
thf(fact_7281_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri6224792872505173163uint32 @ ( numeral_numeral_nat @ K ) @ one_one_uint32 )
      = one_one_uint32 ) ).

% signed_take_bit_numeral_of_1
thf(fact_7282_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_7283_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6224792872505173163uint32 @ N2 @ ( uminus_uminus_uint32 @ one_one_uint32 ) )
      = ( uminus_uminus_uint32 @ one_one_uint32 ) ) ).

% signed_take_bit_of_minus_1
thf(fact_7284_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_7285_sin__pi,axiom,
    ( ( sin_real @ pi )
    = zero_zero_real ) ).

% sin_pi
thf(fact_7286_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7865238048354675525l_num1 @ ( numera7442385471795722001l_num1 @ K ) )
      = ( numera7442385471795722001l_num1 @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_7287_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_7288_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_7289_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_7290_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7865238048354675525l_num1 @ ( uminus8244633308260627903l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) )
      = ( uminus8244633308260627903l_num1 @ ( neg_nu7865238048354675525l_num1 @ ( numera7442385471795722001l_num1 @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7291_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7292_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5314729912787363643uint32 @ ( uminus_uminus_uint32 @ ( numera9087168376688890119uint32 @ K ) ) )
      = ( uminus_uminus_uint32 @ ( neg_nu5314729912787363643uint32 @ ( numera9087168376688890119uint32 @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_7293_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_7294_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_7295_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_7296_divide__le__0__abs__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A3 @ ( abs_abs_real @ B3 ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A3 @ zero_zero_real )
        | ( B3 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_7297_divide__le__0__abs__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A3 @ ( abs_abs_rat @ B3 ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
        | ( B3 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_7298_zero__le__divide__abs__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A3 @ ( abs_abs_real @ B3 ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A3 )
        | ( B3 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_7299_zero__le__divide__abs__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A3 @ ( abs_abs_rat @ B3 ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
        | ( B3 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_7300_abs__of__nonpos,axiom,
    ! [A3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A3 )
        = ( uminus1351360451143612070nteger @ A3 ) ) ) ).

% abs_of_nonpos
thf(fact_7301_abs__of__nonpos,axiom,
    ! [A3: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( abs_abs_real @ A3 )
        = ( uminus_uminus_real @ A3 ) ) ) ).

% abs_of_nonpos
thf(fact_7302_abs__of__nonpos,axiom,
    ! [A3: rat] :
      ( ( ord_less_eq_rat @ A3 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A3 )
        = ( uminus_uminus_rat @ A3 ) ) ) ).

% abs_of_nonpos
thf(fact_7303_abs__of__nonpos,axiom,
    ! [A3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( abs_abs_int @ A3 )
        = ( uminus_uminus_int @ A3 ) ) ) ).

% abs_of_nonpos
thf(fact_7304_artanh__minus__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).

% artanh_minus_real
thf(fact_7305_zero__less__power__abs__iff,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N2 ) )
      = ( ( A3 != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7306_zero__less__power__abs__iff,axiom,
    ! [A3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N2 ) )
      = ( ( A3 != zero_zero_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7307_zero__less__power__abs__iff,axiom,
    ! [A3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N2 ) )
      = ( ( A3 != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7308_zero__less__power__abs__iff,axiom,
    ! [A3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N2 ) )
      = ( ( A3 != zero_zero_int )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_7309_abs__power2,axiom,
    ! [A3: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_7310_abs__power2,axiom,
    ! [A3: int] :
      ( ( abs_abs_int @ ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_7311_abs__power2,axiom,
    ! [A3: real] :
      ( ( abs_abs_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_7312_abs__power2,axiom,
    ! [A3: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_7313_power2__abs,axiom,
    ! [A3: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_7314_power2__abs,axiom,
    ! [A3: int] :
      ( ( power_power_int @ ( abs_abs_int @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_7315_power2__abs,axiom,
    ! [A3: real] :
      ( ( power_power_real @ ( abs_abs_real @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_7316_power2__abs,axiom,
    ! [A3: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_7317_signed__take__bit__Suc__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_7318_sin__cos__squared__add3,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ X2 ) ) @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ X2 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_7319_sin__npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_7320_sin__npi2,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_7321_dbl__simps_I3_J,axiom,
    ( ( neg_nu5314729912787363643uint32 @ one_one_uint32 )
    = ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7322_dbl__simps_I3_J,axiom,
    ( ( neg_nu7865238048354675525l_num1 @ one_on7727431528512463931l_num1 )
    = ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7323_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7324_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7325_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_7326_sin__npi__int,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_7327_power__even__abs__numeral,axiom,
    ! [W: num,A3: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A3 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A3 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_7328_power__even__abs__numeral,axiom,
    ! [W: num,A3: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A3 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A3 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_7329_power__even__abs__numeral,axiom,
    ! [W: num,A3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A3 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A3 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_7330_power__even__abs__numeral,axiom,
    ! [W: num,A3: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A3 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_7331_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_7332_signed__take__bit__0,axiom,
    ! [A3: word_N3645301735248828278l_num1] :
      ( ( bit_ri1375673916561920181l_num1 @ zero_zero_nat @ A3 )
      = ( uminus8244633308260627903l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7333_signed__take__bit__0,axiom,
    ! [A3: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A3 )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7334_signed__take__bit__0,axiom,
    ! [A3: uint32] :
      ( ( bit_ri6224792872505173163uint32 @ zero_zero_nat @ A3 )
      = ( uminus_uminus_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7335_signed__take__bit__0,axiom,
    ! [A3: int] :
      ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A3 )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_7336_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7337_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7338_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7339_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7340_cos__periodic,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X2 ) ) ).

% cos_periodic
thf(fact_7341_sin__periodic,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X2 ) ) ).

% sin_periodic
thf(fact_7342_cos__2pi__minus,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
      = ( cos_real @ X2 ) ) ).

% cos_2pi_minus
thf(fact_7343_sin__cos__squared__add,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_7344_sin__cos__squared__add,axiom,
    ! [X2: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_7345_sin__cos__squared__add2,axiom,
    ! [X2: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_7346_sin__cos__squared__add2,axiom,
    ! [X2: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_7347_signed__take__bit__Suc__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_7348_sin__2npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_7349_cos__2npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_7350_sin__2pi__minus,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X2 ) )
      = ( uminus_uminus_real @ ( sin_real @ X2 ) ) ) ).

% sin_2pi_minus
thf(fact_7351_sin__int__2pin,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_7352_cos__int__2pin,axiom,
    ! [N2: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_7353_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_7354_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_7355_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_7356_cos__npi__int,axiom,
    ! [N2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_7357_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_7358_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_7359_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_7360_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_7361_abs__mult,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_mult
thf(fact_7362_abs__mult,axiom,
    ! [A3: real,B3: real] :
      ( ( abs_abs_real @ ( times_times_real @ A3 @ B3 ) )
      = ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_mult
thf(fact_7363_abs__mult,axiom,
    ! [A3: rat,B3: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A3 @ B3 ) )
      = ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_mult
thf(fact_7364_abs__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( abs_abs_int @ ( times_times_int @ A3 @ B3 ) )
      = ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_mult
thf(fact_7365_power__abs,axiom,
    ! [A3: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A3 @ N2 ) )
      = ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N2 ) ) ).

% power_abs
thf(fact_7366_power__abs,axiom,
    ! [A3: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A3 @ N2 ) )
      = ( power_power_int @ ( abs_abs_int @ A3 ) @ N2 ) ) ).

% power_abs
thf(fact_7367_power__abs,axiom,
    ! [A3: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A3 @ N2 ) )
      = ( power_power_real @ ( abs_abs_real @ A3 ) @ N2 ) ) ).

% power_abs
thf(fact_7368_power__abs,axiom,
    ! [A3: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A3 @ N2 ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N2 ) ) ).

% power_abs
thf(fact_7369_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_7370_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_7371_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_7372_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_7373_sin__zero__abs__cos__one,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_7374_cos__one__sin__zero,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
     => ( ( sin_real @ X2 )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_7375_abs__ge__self,axiom,
    ! [A3: real] : ( ord_less_eq_real @ A3 @ ( abs_abs_real @ A3 ) ) ).

% abs_ge_self
thf(fact_7376_abs__ge__self,axiom,
    ! [A3: code_integer] : ( ord_le3102999989581377725nteger @ A3 @ ( abs_abs_Code_integer @ A3 ) ) ).

% abs_ge_self
thf(fact_7377_abs__ge__self,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ A3 @ ( abs_abs_rat @ A3 ) ) ).

% abs_ge_self
thf(fact_7378_abs__ge__self,axiom,
    ! [A3: int] : ( ord_less_eq_int @ A3 @ ( abs_abs_int @ A3 ) ) ).

% abs_ge_self
thf(fact_7379_abs__le__D1,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
     => ( ord_less_eq_real @ A3 @ B3 ) ) ).

% abs_le_D1
thf(fact_7380_abs__le__D1,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
     => ( ord_le3102999989581377725nteger @ A3 @ B3 ) ) ).

% abs_le_D1
thf(fact_7381_abs__le__D1,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
     => ( ord_less_eq_rat @ A3 @ B3 ) ) ).

% abs_le_D1
thf(fact_7382_abs__le__D1,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
     => ( ord_less_eq_int @ A3 @ B3 ) ) ).

% abs_le_D1
thf(fact_7383_abs__eq__0__iff,axiom,
    ! [A3: code_integer] :
      ( ( ( abs_abs_Code_integer @ A3 )
        = zero_z3403309356797280102nteger )
      = ( A3 = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_7384_abs__eq__0__iff,axiom,
    ! [A3: real] :
      ( ( ( abs_abs_real @ A3 )
        = zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_7385_abs__eq__0__iff,axiom,
    ! [A3: rat] :
      ( ( ( abs_abs_rat @ A3 )
        = zero_zero_rat )
      = ( A3 = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_7386_abs__eq__0__iff,axiom,
    ! [A3: int] :
      ( ( ( abs_abs_int @ A3 )
        = zero_zero_int )
      = ( A3 = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_7387_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).

% signed_take_bit_mult
thf(fact_7388_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L2 ) ) ) ).

% signed_take_bit_add
thf(fact_7389_signed__take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_7390_abs__eq__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ( abs_abs_Code_integer @ X2 )
        = ( abs_abs_Code_integer @ Y2 ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus1351360451143612070nteger @ Y2 ) ) ) ) ).

% abs_eq_iff
thf(fact_7391_abs__eq__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( abs_abs_real @ X2 )
        = ( abs_abs_real @ Y2 ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_real @ Y2 ) ) ) ) ).

% abs_eq_iff
thf(fact_7392_abs__eq__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ( abs_abs_rat @ X2 )
        = ( abs_abs_rat @ Y2 ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_rat @ Y2 ) ) ) ) ).

% abs_eq_iff
thf(fact_7393_abs__eq__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ( abs_abs_int @ X2 )
        = ( abs_abs_int @ Y2 ) )
      = ( ( X2 = Y2 )
        | ( X2
          = ( uminus_uminus_int @ Y2 ) ) ) ) ).

% abs_eq_iff
thf(fact_7394_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% signed_take_bit_diff
thf(fact_7395_sin__zero__norm__cos__one,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7396_sin__zero__norm__cos__one,axiom,
    ! [X2: complex] :
      ( ( ( sin_complex @ X2 )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X2 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7397_sin__zero__pi__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ pi )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_7398_sin__double,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X2 ) ) @ ( cos_real @ X2 ) ) ) ).

% sin_double
thf(fact_7399_sincos__principal__value,axiom,
    ! [X2: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X2 ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X2 ) ) ) ).

% sincos_principal_value
thf(fact_7400_abs__ge__zero,axiom,
    ! [A3: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A3 ) ) ).

% abs_ge_zero
thf(fact_7401_abs__ge__zero,axiom,
    ! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A3 ) ) ).

% abs_ge_zero
thf(fact_7402_abs__ge__zero,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A3 ) ) ).

% abs_ge_zero
thf(fact_7403_abs__ge__zero,axiom,
    ! [A3: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A3 ) ) ).

% abs_ge_zero
thf(fact_7404_abs__of__pos,axiom,
    ! [A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A3 )
     => ( ( abs_abs_Code_integer @ A3 )
        = A3 ) ) ).

% abs_of_pos
thf(fact_7405_abs__of__pos,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( abs_abs_real @ A3 )
        = A3 ) ) ).

% abs_of_pos
thf(fact_7406_abs__of__pos,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A3 )
     => ( ( abs_abs_rat @ A3 )
        = A3 ) ) ).

% abs_of_pos
thf(fact_7407_abs__of__pos,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ( abs_abs_int @ A3 )
        = A3 ) ) ).

% abs_of_pos
thf(fact_7408_abs__not__less__zero,axiom,
    ! [A3: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_7409_abs__not__less__zero,axiom,
    ! [A3: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A3 ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7410_abs__not__less__zero,axiom,
    ! [A3: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_7411_abs__not__less__zero,axiom,
    ! [A3: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A3 ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7412_abs__triangle__ineq,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_7413_abs__triangle__ineq,axiom,
    ! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_7414_abs__triangle__ineq,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A3 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_7415_abs__triangle__ineq,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A3 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_triangle_ineq
thf(fact_7416_abs__mult__less,axiom,
    ! [A3: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B3 ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7417_abs__mult__less,axiom,
    ! [A3: real,C: real,B3: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A3 ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B3 ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7418_abs__mult__less,axiom,
    ! [A3: rat,C: rat,B3: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B3 ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7419_abs__mult__less,axiom,
    ! [A3: int,C: int,B3: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A3 ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7420_abs__triangle__ineq2__sym,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A3 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7421_abs__triangle__ineq2__sym,axiom,
    ! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ A3 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7422_abs__triangle__ineq2__sym,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A3 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7423_abs__triangle__ineq2__sym,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ A3 ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7424_abs__triangle__ineq3,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7425_abs__triangle__ineq3,axiom,
    ! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7426_abs__triangle__ineq3,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7427_abs__triangle__ineq3,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq3
thf(fact_7428_abs__triangle__ineq2,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7429_abs__triangle__ineq2,axiom,
    ! [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7430_abs__triangle__ineq2,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7431_abs__triangle__ineq2,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) ) ).

% abs_triangle_ineq2
thf(fact_7432_nonzero__abs__divide,axiom,
    ! [B3: real,A3: real] :
      ( ( B3 != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A3 @ B3 ) )
        = ( divide_divide_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7433_nonzero__abs__divide,axiom,
    ! [B3: rat,A3: rat] :
      ( ( B3 != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A3 @ B3 ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7434_abs__leI,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_7435_abs__leI,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 )
       => ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_7436_abs__leI,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ A3 @ B3 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_7437_abs__leI,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ B3 )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 )
       => ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 ) ) ) ).

% abs_leI
thf(fact_7438_abs__le__D2,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_7439_abs__le__D2,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_7440_abs__le__D2,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_7441_abs__le__D2,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ).

% abs_le_D2
thf(fact_7442_abs__le__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
      = ( ( ord_le3102999989581377725nteger @ A3 @ B3 )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_7443_abs__le__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ B3 )
      = ( ( ord_less_eq_real @ A3 @ B3 )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_7444_abs__le__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ B3 )
      = ( ( ord_less_eq_rat @ A3 @ B3 )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_7445_abs__le__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ B3 )
      = ( ( ord_less_eq_int @ A3 @ B3 )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).

% abs_le_iff
thf(fact_7446_abs__ge__minus__self,axiom,
    ! [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A3 ) @ ( abs_abs_Code_integer @ A3 ) ) ).

% abs_ge_minus_self
thf(fact_7447_abs__ge__minus__self,axiom,
    ! [A3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A3 ) @ ( abs_abs_real @ A3 ) ) ).

% abs_ge_minus_self
thf(fact_7448_abs__ge__minus__self,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A3 ) @ ( abs_abs_rat @ A3 ) ) ).

% abs_ge_minus_self
thf(fact_7449_abs__ge__minus__self,axiom,
    ! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A3 ) @ ( abs_abs_int @ A3 ) ) ).

% abs_ge_minus_self
thf(fact_7450_abs__less__iff,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A3 ) @ B3 )
      = ( ( ord_le6747313008572928689nteger @ A3 @ B3 )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A3 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_7451_abs__less__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A3 ) @ B3 )
      = ( ( ord_less_real @ A3 @ B3 )
        & ( ord_less_real @ ( uminus_uminus_real @ A3 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_7452_abs__less__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A3 ) @ B3 )
      = ( ( ord_less_rat @ A3 @ B3 )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A3 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_7453_abs__less__iff,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A3 ) @ B3 )
      = ( ( ord_less_int @ A3 @ B3 )
        & ( ord_less_int @ ( uminus_uminus_int @ A3 ) @ B3 ) ) ) ).

% abs_less_iff
thf(fact_7454_sin__x__le__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).

% sin_x_le_x
thf(fact_7455_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).

% dbl_def
thf(fact_7456_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).

% dbl_def
thf(fact_7457_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).

% dbl_def
thf(fact_7458_cos__squared__eq,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7459_cos__squared__eq,axiom,
    ! [X2: real] :
      ( ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7460_sin__squared__eq,axiom,
    ! [X2: complex] :
      ( ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7461_sin__squared__eq,axiom,
    ! [X2: real] :
      ( ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7462_dense__eq0__I,axiom,
    ! [X2: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E ) )
     => ( X2 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_7463_dense__eq0__I,axiom,
    ! [X2: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E ) )
     => ( X2 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_7464_abs__mult__pos,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y2 ) @ X2 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_7465_abs__mult__pos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ ( abs_abs_real @ Y2 ) @ X2 )
        = ( abs_abs_real @ ( times_times_real @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_7466_abs__mult__pos,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y2 ) @ X2 )
        = ( abs_abs_rat @ ( times_times_rat @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_7467_abs__mult__pos,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( times_times_int @ ( abs_abs_int @ Y2 ) @ X2 )
        = ( abs_abs_int @ ( times_times_int @ Y2 @ X2 ) ) ) ) ).

% abs_mult_pos
thf(fact_7468_abs__eq__mult,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
          | ( ord_le3102999989581377725nteger @ A3 @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
          | ( ord_le3102999989581377725nteger @ B3 @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A3 @ B3 ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_7469_abs__eq__mult,axiom,
    ! [A3: real,B3: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
          | ( ord_less_eq_real @ A3 @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B3 )
          | ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A3 @ B3 ) )
        = ( times_times_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_7470_abs__eq__mult,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
          | ( ord_less_eq_rat @ A3 @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
          | ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A3 @ B3 ) )
        = ( times_times_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_7471_abs__eq__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A3 )
          | ( ord_less_eq_int @ A3 @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B3 )
          | ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A3 @ B3 ) )
        = ( times_times_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ) ).

% abs_eq_mult
thf(fact_7472_abs__minus__le__zero,axiom,
    ! [A3: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A3 ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_7473_abs__minus__le__zero,axiom,
    ! [A3: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A3 ) ) @ zero_zero_real ) ).

% abs_minus_le_zero
thf(fact_7474_abs__minus__le__zero,axiom,
    ! [A3: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A3 ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_7475_abs__minus__le__zero,axiom,
    ! [A3: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A3 ) ) @ zero_zero_int ) ).

% abs_minus_le_zero
thf(fact_7476_eq__abs__iff_H,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( A3
        = ( abs_abs_Code_integer @ B3 ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A3 )
        & ( ( B3 = A3 )
          | ( B3
            = ( uminus1351360451143612070nteger @ A3 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7477_eq__abs__iff_H,axiom,
    ! [A3: real,B3: real] :
      ( ( A3
        = ( abs_abs_real @ B3 ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A3 )
        & ( ( B3 = A3 )
          | ( B3
            = ( uminus_uminus_real @ A3 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7478_eq__abs__iff_H,axiom,
    ! [A3: rat,B3: rat] :
      ( ( A3
        = ( abs_abs_rat @ B3 ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A3 )
        & ( ( B3 = A3 )
          | ( B3
            = ( uminus_uminus_rat @ A3 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7479_eq__abs__iff_H,axiom,
    ! [A3: int,B3: int] :
      ( ( A3
        = ( abs_abs_int @ B3 ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A3 )
        & ( ( B3 = A3 )
          | ( B3
            = ( uminus_uminus_int @ A3 ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7480_abs__eq__iff_H,axiom,
    ! [A3: code_integer,B3: code_integer] :
      ( ( ( abs_abs_Code_integer @ A3 )
        = B3 )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
        & ( ( A3 = B3 )
          | ( A3
            = ( uminus1351360451143612070nteger @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7481_abs__eq__iff_H,axiom,
    ! [A3: real,B3: real] :
      ( ( ( abs_abs_real @ A3 )
        = B3 )
      = ( ( ord_less_eq_real @ zero_zero_real @ B3 )
        & ( ( A3 = B3 )
          | ( A3
            = ( uminus_uminus_real @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7482_abs__eq__iff_H,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ( abs_abs_rat @ A3 )
        = B3 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
        & ( ( A3 = B3 )
          | ( A3
            = ( uminus_uminus_rat @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7483_abs__eq__iff_H,axiom,
    ! [A3: int,B3: int] :
      ( ( ( abs_abs_int @ A3 )
        = B3 )
      = ( ( ord_less_eq_int @ zero_zero_int @ B3 )
        & ( ( A3 = B3 )
          | ( A3
            = ( uminus_uminus_int @ B3 ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7484_abs__div__pos,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y2 )
        = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y2 ) ) ) ) ).

% abs_div_pos
thf(fact_7485_abs__div__pos,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y2 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y2 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y2 ) ) ) ) ).

% abs_div_pos
thf(fact_7486_zero__le__power__abs,axiom,
    ! [A3: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7487_zero__le__power__abs,axiom,
    ! [A3: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7488_zero__le__power__abs,axiom,
    ! [A3: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7489_zero__le__power__abs,axiom,
    ! [A3: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A3 ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7490_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_7491_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_7492_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_7493_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if
thf(fact_7494_abs__of__neg,axiom,
    ! [A3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A3 )
        = ( uminus1351360451143612070nteger @ A3 ) ) ) ).

% abs_of_neg
thf(fact_7495_abs__of__neg,axiom,
    ! [A3: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( abs_abs_real @ A3 )
        = ( uminus_uminus_real @ A3 ) ) ) ).

% abs_of_neg
thf(fact_7496_abs__of__neg,axiom,
    ! [A3: rat] :
      ( ( ord_less_rat @ A3 @ zero_zero_rat )
     => ( ( abs_abs_rat @ A3 )
        = ( uminus_uminus_rat @ A3 ) ) ) ).

% abs_of_neg
thf(fact_7497_abs__of__neg,axiom,
    ! [A3: int] :
      ( ( ord_less_int @ A3 @ zero_zero_int )
     => ( ( abs_abs_int @ A3 )
        = ( uminus_uminus_int @ A3 ) ) ) ).

% abs_of_neg
thf(fact_7498_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_7499_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_7500_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_7501_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A2: int] : ( if_int @ ( ord_less_int @ A2 @ zero_zero_int ) @ ( uminus_uminus_int @ A2 ) @ A2 ) ) ) ).

% abs_if_raw
thf(fact_7502_abs__diff__le__iff,axiom,
    ! [X2: code_integer,A3: code_integer,R: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A3 ) ) @ R )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A3 @ R ) @ X2 )
        & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A3 @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7503_abs__diff__le__iff,axiom,
    ! [X2: real,A3: real,R: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A3 @ R ) @ X2 )
        & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A3 @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7504_abs__diff__le__iff,axiom,
    ! [X2: rat,A3: rat,R: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ R ) @ X2 )
        & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A3 @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7505_abs__diff__le__iff,axiom,
    ! [X2: int,A3: int,R: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A3 @ R ) @ X2 )
        & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A3 @ R ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7506_abs__diff__triangle__ineq,axiom,
    ! [A3: code_integer,B3: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A3 @ B3 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7507_abs__diff__triangle__ineq,axiom,
    ! [A3: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A3 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A3 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7508_abs__diff__triangle__ineq,axiom,
    ! [A3: rat,B3: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A3 @ B3 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7509_abs__diff__triangle__ineq,axiom,
    ! [A3: int,B3: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A3 @ B3 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7510_abs__triangle__ineq4,axiom,
    ! [A3: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A3 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7511_abs__triangle__ineq4,axiom,
    ! [A3: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A3 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7512_abs__triangle__ineq4,axiom,
    ! [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A3 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7513_abs__triangle__ineq4,axiom,
    ! [A3: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A3 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) ) ) ).

% abs_triangle_ineq4
thf(fact_7514_abs__diff__less__iff,axiom,
    ! [X2: code_integer,A3: code_integer,R: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A3 ) ) @ R )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A3 @ R ) @ X2 )
        & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A3 @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7515_abs__diff__less__iff,axiom,
    ! [X2: real,A3: real,R: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_real @ ( minus_minus_real @ A3 @ R ) @ X2 )
        & ( ord_less_real @ X2 @ ( plus_plus_real @ A3 @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7516_abs__diff__less__iff,axiom,
    ! [X2: rat,A3: rat,R: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A3 @ R ) @ X2 )
        & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A3 @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7517_abs__diff__less__iff,axiom,
    ! [X2: int,A3: int,R: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A3 ) ) @ R )
      = ( ( ord_less_int @ ( minus_minus_int @ A3 @ R ) @ X2 )
        & ( ord_less_int @ X2 @ ( plus_plus_int @ A3 @ R ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7518_sin__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero
thf(fact_7519_sin__x__ge__neg__x,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7520_sin__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_ge_zero
thf(fact_7521_cos__inj__pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ( cos_real @ X2 )
                = ( cos_real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_7522_cos__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) )
              = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_7523_cos__monotone__0__pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_7524_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7525_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7526_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7527_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7528_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7529_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7530_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A2: real] : ( if_real @ ( ord_less_real @ A2 @ zero_zero_real ) @ ( uminus_uminus_real @ A2 ) @ A2 ) ) ) ).

% abs_real_def
thf(fact_7531_cos__double,axiom,
    ! [X2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7532_cos__double,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7533_lemma__interval__lt,axiom,
    ! [A3: real,X2: real,B3: real] :
      ( ( ord_less_real @ A3 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D2 )
               => ( ( ord_less_real @ A3 @ Y4 )
                  & ( ord_less_real @ Y4 @ B3 ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7534_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_7535_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_7536_abs__add__one__gt__zero,axiom,
    ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7537_abs__add__one__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7538_abs__add__one__gt__zero,axiom,
    ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7539_abs__add__one__gt__zero,axiom,
    ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7540_of__int__leD,axiom,
    ! [N2: int,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_leD
thf(fact_7541_of__int__leD,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).

% of_int_leD
thf(fact_7542_of__int__leD,axiom,
    ! [N2: int,X2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_leD
thf(fact_7543_of__int__leD,axiom,
    ! [N2: int,X2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).

% of_int_leD
thf(fact_7544_of__int__lessD,axiom,
    ! [N2: int,X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7545_of__int__lessD,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7546_of__int__lessD,axiom,
    ! [N2: int,X2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7547_of__int__lessD,axiom,
    ! [N2: int,X2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X2 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X2 ) ) ) ).

% of_int_lessD
thf(fact_7548_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7549_cos__monotone__0__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_7550_cos__mono__less__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) )
              = ( ord_less_real @ Y2 @ X2 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_7551_sin__eq__0__pi,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
     => ( ( ord_less_real @ X2 @ pi )
       => ( ( ( sin_real @ X2 )
            = zero_zero_real )
         => ( X2 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_7552_cos__monotone__minus__pi__0_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y2 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_7553_lemma__interval,axiom,
    ! [A3: real,X2: real,B3: real] :
      ( ( ord_less_real @ A3 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D2 )
               => ( ( ord_less_eq_real @ A3 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B3 ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7554_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_7555_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_7556_sin__zero__iff__int2,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( X2
            = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_7557_sincos__total__pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ pi )
            & ( X2
              = ( cos_real @ T3 ) )
            & ( Y2
              = ( sin_real @ T3 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7558_sin__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7559_abs__le__square__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y2 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7560_abs__le__square__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y2 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7561_abs__le__square__iff,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y2 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7562_abs__le__square__iff,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y2 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7563_abs__square__eq__1,axiom,
    ! [X2: rat] :
      ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X2 )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_7564_abs__square__eq__1,axiom,
    ! [X2: int] :
      ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X2 )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_7565_abs__square__eq__1,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X2 )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_7566_abs__square__eq__1,axiom,
    ! [X2: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X2 )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_7567_power__even__abs,axiom,
    ! [N2: nat,A3: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( abs_abs_rat @ A3 ) @ N2 )
        = ( power_power_rat @ A3 @ N2 ) ) ) ).

% power_even_abs
thf(fact_7568_power__even__abs,axiom,
    ! [N2: nat,A3: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( abs_abs_int @ A3 ) @ N2 )
        = ( power_power_int @ A3 @ N2 ) ) ) ).

% power_even_abs
thf(fact_7569_power__even__abs,axiom,
    ! [N2: nat,A3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( abs_abs_real @ A3 ) @ N2 )
        = ( power_power_real @ A3 @ N2 ) ) ) ).

% power_even_abs
thf(fact_7570_power__even__abs,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A3 ) @ N2 )
        = ( power_8256067586552552935nteger @ A3 @ N2 ) ) ) ).

% power_even_abs
thf(fact_7571_cos__expansion__lemma,axiom,
    ! [X2: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X2 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7572_sin__gt__zero__02,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_7573_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7574_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_7575_cos__is__zero,axiom,
    ? [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
      & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X3 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X3 ) ) ) ).

% cos_is_zero
thf(fact_7576_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_7577_cos__monotone__minus__pi__0,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y2 ) @ ( cos_real @ X2 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_7578_cos__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X3 )
            & ( ord_less_eq_real @ X3 @ pi )
            & ( ( cos_real @ X3 )
              = Y2 )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y2 ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% cos_total
thf(fact_7579_even__signed__take__bit__iff,axiom,
    ! [M: nat,A3: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ M @ A3 ) )
      = ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_signed_take_bit_iff
thf(fact_7580_even__signed__take__bit__iff,axiom,
    ! [M: nat,A3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A3 ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_signed_take_bit_iff
thf(fact_7581_even__signed__take__bit__iff,axiom,
    ! [M: nat,A3: word_N3645301735248828278l_num1] :
      ( ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ M @ A3 ) )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_signed_take_bit_iff
thf(fact_7582_even__signed__take__bit__iff,axiom,
    ! [M: nat,A3: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A3 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ).

% even_signed_take_bit_iff
thf(fact_7583_sincos__total__pi__half,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X2
                = ( cos_real @ T3 ) )
              & ( Y2
                = ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7584_sincos__total__2pi__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X2
            = ( cos_real @ T3 ) )
          & ( Y2
            = ( sin_real @ T3 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7585_power2__le__iff__abs__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7586_power2__le__iff__abs__le,axiom,
    ! [Y2: code_integer,X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7587_power2__le__iff__abs__le,axiom,
    ! [Y2: rat,X2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y2 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7588_power2__le__iff__abs__le,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y2 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7589_abs__square__le__1,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_7590_abs__square__le__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_7591_abs__square__le__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_7592_abs__square__le__1,axiom,
    ! [X2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_7593_abs__square__less__1,axiom,
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_7594_abs__square__less__1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_7595_abs__square__less__1,axiom,
    ! [X2: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_7596_abs__square__less__1,axiom,
    ! [X2: int] :
      ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_7597_power__mono__even,axiom,
    ! [N2: nat,A3: real,B3: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A3 ) @ ( abs_abs_real @ B3 ) )
       => ( ord_less_eq_real @ ( power_power_real @ A3 @ N2 ) @ ( power_power_real @ B3 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7598_power__mono__even,axiom,
    ! [N2: nat,A3: code_integer,B3: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A3 ) @ ( abs_abs_Code_integer @ B3 ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A3 @ N2 ) @ ( power_8256067586552552935nteger @ B3 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7599_power__mono__even,axiom,
    ! [N2: nat,A3: rat,B3: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A3 ) @ ( abs_abs_rat @ B3 ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A3 @ N2 ) @ ( power_power_rat @ B3 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7600_power__mono__even,axiom,
    ! [N2: nat,A3: int,B3: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A3 ) @ ( abs_abs_int @ B3 ) )
       => ( ord_less_eq_int @ ( power_power_int @ A3 @ N2 ) @ ( power_power_int @ B3 @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7601_sincos__total__2pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X2
                  = ( cos_real @ T3 ) )
               => ( Y2
                 != ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7602_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_7603_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_7604_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_7605_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_7606_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_7607_sin__pi__divide__n__ge__0,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_7608_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7609_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7610_sin__gt__zero2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).

% sin_gt_zero2
thf(fact_7611_sin__lt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_7612_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_7613_cos__double__less__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_7614_signed__take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_7615_cos__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero
thf(fact_7616_sin__inj__pi,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X2 )
                = ( sin_real @ Y2 ) )
             => ( X2 = Y2 ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_7617_sin__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) )
              = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_7618_sin__monotone__2pi__le,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y2 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_7619_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_7620_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_7621_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_7622_cos__one__2pi__int,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
      = ( ? [X: int] :
            ( X2
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_7623_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_7624_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_7625_cos__treble__cos,axiom,
    ! [X2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X2 ) ) ) ) ).

% cos_treble_cos
thf(fact_7626_cos__treble__cos,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X2 ) ) ) ) ).

% cos_treble_cos
thf(fact_7627_sin__le__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ pi @ X2 )
     => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_7628_sin__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_7629_sin__mono__less__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y2 ) )
              = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_7630_sin__monotone__2pi,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y2 ) @ ( sin_real @ X2 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_7631_sin__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ? [X3: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X3 )
              = Y2 )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y2 ) )
               => ( Y4 = X3 ) ) ) ) ) ).

% sin_total
thf(fact_7632_cos__gt__zero__pi,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_7633_cos__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).

% cos_ge_zero
thf(fact_7634_cos__one__2pi,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = one_one_real )
      = ( ? [X: nat] :
            ( X2
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X2
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7635_of__int__round__abs__le,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7636_of__int__round__abs__le,axiom,
    ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7637_round__unique_H,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X2 )
        = N2 ) ) ).

% round_unique'
thf(fact_7638_round__unique_H,axiom,
    ! [X2: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X2 )
        = N2 ) ) ).

% round_unique'
thf(fact_7639_sin__pi__divide__n__gt__0,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% sin_pi_divide_n_gt_0
thf(fact_7640_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_7641_signed__take__bit__Suc,axiom,
    ! [N2: nat,A3: word_N3645301735248828278l_num1] :
      ( ( bit_ri1375673916561920181l_num1 @ ( suc @ N2 ) @ A3 )
      = ( plus_p361126936061061375l_num1 @ ( modulo1504961113040953224l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) @ ( times_7065122842183080059l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( bit_ri1375673916561920181l_num1 @ N2 @ ( divide1791077408188789448l_num1 @ A3 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7642_signed__take__bit__Suc,axiom,
    ! [N2: nat,A3: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A3 )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7643_signed__take__bit__Suc,axiom,
    ! [N2: nat,A3: uint32] :
      ( ( bit_ri6224792872505173163uint32 @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_uint32 @ ( modulo_modulo_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) @ ( times_times_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ ( bit_ri6224792872505173163uint32 @ N2 @ ( divide_divide_uint32 @ A3 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7644_signed__take__bit__Suc,axiom,
    ! [N2: nat,A3: int] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A3 )
      = ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_7645_sin__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_7646_cos__zero__iff__int,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = zero_zero_real )
      = ( ? [I4: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
            & ( X2
              = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_7647_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_7648_sin__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sin_real @ X2 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_7649_sin__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ X2 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7650_cos__zero__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( cos_real @ X2 )
          = zero_zero_real )
       => ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_7651_cos__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X2
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7652_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_7653_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X2: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7654_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X2: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7655_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X2: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7656_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X2: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7657_monoseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7658_arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( arctan @ X2 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7659_summable__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7660_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_7661_zdvd1__eq,axiom,
    ! [X2: int] :
      ( ( dvd_dvd_int @ X2 @ one_one_int )
      = ( ( abs_abs_int @ X2 )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_7662_arctan__zero__zero,axiom,
    ( ( arctan @ zero_zero_real )
    = zero_zero_real ) ).

% arctan_zero_zero
thf(fact_7663_arctan__eq__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( arctan @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% arctan_eq_zero_iff
thf(fact_7664_summable__zero,axiom,
    ( summable_complex
    @ ^ [N: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_7665_summable__zero,axiom,
    ( summable_real
    @ ^ [N: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_7666_summable__zero,axiom,
    ( summable_nat
    @ ^ [N: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_7667_summable__zero,axiom,
    ( summable_int
    @ ^ [N: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_7668_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R6: nat] : ( if_complex @ ( R6 = I ) @ ( F @ R6 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_7669_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R6: nat] : ( if_real @ ( R6 = I ) @ ( F @ R6 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_7670_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R6: nat] : ( if_nat @ ( R6 = I ) @ ( F @ R6 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_7671_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R6: nat] : ( if_int @ ( R6 = I ) @ ( F @ R6 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_7672_summable__iff__shift,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7673_summable__iff__shift,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
      = ( summable_complex @ F ) ) ).

% summable_iff_shift
thf(fact_7674_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_7675_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_7676_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_7677_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_7678_arctan__less__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_7679_zero__less__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% zero_less_arctan_iff
thf(fact_7680_arctan__le__zero__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_7681_zero__le__arctan__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_arctan_iff
thf(fact_7682_summable__cmult__iff,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7683_summable__cmult__iff,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_cmult_iff
thf(fact_7684_summable__divide__iff,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_complex )
        | ( summable_complex @ F ) ) ) ).

% summable_divide_iff
thf(fact_7685_summable__divide__iff,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
      = ( ( C = zero_zero_real )
        | ( summable_real @ F ) ) ) ).

% summable_divide_iff
thf(fact_7686_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_7687_less__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% less_numeral_Suc
thf(fact_7688_less__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% less_Suc_numeral
thf(fact_7689_le__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% le_numeral_Suc
thf(fact_7690_le__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% le_Suc_numeral
thf(fact_7691_diff__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).

% diff_Suc_numeral
thf(fact_7692_diff__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).

% diff_numeral_Suc
thf(fact_7693_max__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% max_Suc_numeral
thf(fact_7694_max__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% max_numeral_Suc
thf(fact_7695_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7696_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_7697_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_7698_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_7699_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_7700_summable__norm__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
     => ( summable_real @ F ) ) ).

% summable_norm_cancel
thf(fact_7701_summable__norm__cancel,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
     => ( summable_complex @ F ) ) ).

% summable_norm_cancel
thf(fact_7702_arctan__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% arctan_less_iff
thf(fact_7703_arctan__monotone,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) ) ) ).

% arctan_monotone
thf(fact_7704_summable__comparison__test_H,axiom,
    ! [G: nat > real,N3: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7705_summable__comparison__test_H,axiom,
    ! [G: nat > real,N3: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7706_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N9: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N9 @ N4 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7707_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N9: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N9 @ N4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7708_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_7709_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_7710_summable__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7711_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7712_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7713_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_add
thf(fact_7714_summable__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ).

% summable_mult2
thf(fact_7715_summable__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ).

% summable_mult2
thf(fact_7716_summable__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) ) ) ).

% summable_mult
thf(fact_7717_summable__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) ) ) ).

% summable_mult
thf(fact_7718_summable__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( summable_complex
          @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_diff
thf(fact_7719_summable__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).

% summable_diff
thf(fact_7720_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7721_summable__Suc__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_complex @ F ) ) ).

% summable_Suc_iff
thf(fact_7722_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_7723_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_7724_summable__minus__iff,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
      = ( summable_complex @ F ) ) ).

% summable_minus_iff
thf(fact_7725_summable__minus__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_minus_iff
thf(fact_7726_summable__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) ) ) ).

% summable_minus
thf(fact_7727_summable__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) ) ) ).

% summable_minus
thf(fact_7728_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7729_summable__ignore__initial__segment,axiom,
    ! [F: nat > complex,K: nat] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7730_summable__rabs__cancel,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
     => ( summable_real @ F ) ) ).

% summable_rabs_cancel
thf(fact_7731_powser__insidea,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7732_powser__insidea,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7733_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7734_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7735_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7736_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7737_summable__mult__D,axiom,
    ! [C: complex,F: nat > complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_complex )
       => ( summable_complex @ F ) ) ) ).

% summable_mult_D
thf(fact_7738_summable__mult__D,axiom,
    ! [C: real,F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
     => ( ( C != zero_zero_real )
       => ( summable_real @ F ) ) ) ).

% summable_mult_D
thf(fact_7739_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7740_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7741_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7742_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_7743_abs__div,axiom,
    ! [Y2: int,X2: int] :
      ( ( dvd_dvd_int @ Y2 @ X2 )
     => ( ( abs_abs_int @ ( divide_divide_int @ X2 @ Y2 ) )
        = ( divide_divide_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y2 ) ) ) ) ).

% abs_div
thf(fact_7744_suminf__add,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( plus_plus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7745_suminf__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7746_suminf__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
          = ( suminf_nat
            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7747_suminf__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
          = ( suminf_int
            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_add
thf(fact_7748_suminf__mult,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
        = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).

% suminf_mult
thf(fact_7749_suminf__mult,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
        = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).

% suminf_mult
thf(fact_7750_suminf__mult2,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
        = ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7751_suminf__mult2,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( times_times_real @ ( suminf_real @ F ) @ C )
        = ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ) ).

% suminf_mult2
thf(fact_7752_suminf__diff,axiom,
    ! [F: nat > complex,G: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( summable_complex @ G )
       => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
          = ( suminf_complex
            @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7753_suminf__diff,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
          = ( suminf_real
            @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).

% suminf_diff
thf(fact_7754_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7755_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_7756_suminf__minus,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
        = ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).

% suminf_minus
thf(fact_7757_suminf__minus,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
        = ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_minus
thf(fact_7758_cos__arctan__not__zero,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( arctan @ X2 ) )
     != zero_zero_real ) ).

% cos_arctan_not_zero
thf(fact_7759_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7760_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7761_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7762_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7763_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7764_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N: nat] :
                ( ( F @ N )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7765_suminf__pos,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_pos
thf(fact_7766_suminf__pos,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_pos
thf(fact_7767_suminf__pos,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_pos
thf(fact_7768_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_zero_power'
thf(fact_7769_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_zero_power'
thf(fact_7770_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).

% summable_zero_power'
thf(fact_7771_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).

% summable_0_powser
thf(fact_7772_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).

% summable_0_powser
thf(fact_7773_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7774_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_7775_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7776_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_7777_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7778_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7779_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).

% zabs_def
thf(fact_7780_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_7781_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_7782_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N9: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N9 @ N4 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7783_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N9: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N9 @ N4 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7784_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_7785_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_7786_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs
thf(fact_7787_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7788_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7789_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7790_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7791_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7792_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7793_powser__inside,axiom,
    ! [F: nat > real,X2: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_7794_powser__inside,axiom,
    ! [F: nat > complex,X2: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X2 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_7795_suminf__split__head,axiom,
    ! [F: nat > complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7796_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_7797_complete__algebra__summable__geometric,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7798_complete__algebra__summable__geometric,axiom,
    ! [X2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_7799_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_7800_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_7801_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7802_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_7803_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_7804_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_7805_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_7806_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7807_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_7808_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7809_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_7810_monoseq__realpow,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).

% monoseq_realpow
thf(fact_7811_suminf__exist__split,axiom,
    ! [R: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( summable_real @ F )
       => ? [N10: nat] :
          ! [N5: nat] :
            ( ( ord_less_eq_nat @ N10 @ N5 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7812_suminf__exist__split,axiom,
    ! [R: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R )
     => ( ( summable_complex @ F )
       => ? [N10: nat] :
          ! [N5: nat] :
            ( ( ord_less_eq_nat @ N10 @ N5 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
              @ R ) ) ) ) ).

% suminf_exist_split
thf(fact_7813_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
     => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_7814_Abel__lemma,axiom,
    ! [R: real,R0: real,A3: nat > complex,M8: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R )
     => ( ( ord_less_real @ R @ R0 )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A3 @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M8 )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A3 @ N ) ) @ ( power_power_real @ R @ N ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_7815_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ( ord_less_eq_nat @ M @ I2 )
            & ( ord_less_nat @ I2 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I2: nat] :
                ( ( ord_less_eq_nat @ M @ I2 )
                & ( ord_less_eq_nat @ I2 @ N2 )
                & ( ( F @ I2 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7816_decr__lemma,axiom,
    ! [D: int,X2: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_7817_incr__lemma,axiom,
    ! [D: int,Z: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7818_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_7819_summable__ratio__test,axiom,
    ! [C: real,N3: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N4: nat] :
            ( ( ord_less_eq_nat @ N3 @ N4 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_7820_arctan__ubound,axiom,
    ! [Y2: real] : ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7821_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7822_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7823_arctan__lbound,axiom,
    ! [Y2: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) ) ).

% arctan_lbound
thf(fact_7824_arctan__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_7825_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I2: nat] :
              ( ( ord_less_eq_nat @ I2 @ N2 )
              & ( ( F @ I2 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7826_arctan__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y2 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y2 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_7827_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_7828_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_7829_arctan__double,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_7830_tan__double,axiom,
    ! [X2: complex] :
      ( ( ( cos_complex @ X2 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X2 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7831_tan__double,axiom,
    ! [X2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X2 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7832_even__word__def,axiom,
    ( even_w9054469088133485505l_num1
    = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) ) ) ).

% even_word_def
thf(fact_7833_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_7834_exE__realizer,axiom,
    ! [P: nat > nat > $o,P2: product_prod_nat_nat,Q: product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat] :
      ( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc2626176000494625587at_nat @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_7835_exE__realizer,axiom,
    ! [P: nat > nat > $o,P2: product_prod_nat_nat,Q: $o > $o,F: nat > nat > $o] :
      ( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
     => ( ! [X3: nat,Y3: nat] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc6081775807080527818_nat_o @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_7836_exE__realizer,axiom,
    ! [P: int > int > $o,P2: product_prod_int_int,Q: product_prod_int_int > $o,F: int > int > product_prod_int_int] :
      ( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc4245557441103728435nt_int @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_7837_exE__realizer,axiom,
    ! [P: int > int > $o,P2: product_prod_int_int,Q: $o > $o,F: int > int > $o] :
      ( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc4947309494688390418_int_o @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_7838_exE__realizer,axiom,
    ! [P: int > int > $o,P2: product_prod_int_int,Q: int > $o,F: int > int > int] :
      ( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
     => ( ! [X3: int,Y3: int] :
            ( ( P @ Y3 @ X3 )
           => ( Q @ ( F @ X3 @ Y3 ) ) )
       => ( Q @ ( produc8211389475949308722nt_int @ F @ P2 ) ) ) ) ).

% exE_realizer
thf(fact_7839_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ) > $o] :
      ( ( comple1745167176254620304_VEBTi @ ( partia7782936097874681665Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi )
        @ ^ [Vebt_inserti: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] : ( P @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti ) ) ) )
     => ( ( P
          @ ^ [Vebt_inserti: vEBT_VEBT,T: vEBT_VEBTi,Ti3: nat] :
              ( heap_T1489671670754571048_VEBTi
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) )
       => ( ! [F5: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBT,A2: vEBT_VEBTi,B2: nat] :
                    ( vEBT_c6028912655521741485_VEBTi
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ B2 @ B2 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                        @ ^ [Minma: product_prod_nat_nat] :
                            ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                            @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ X8 ) )
                              @ ^ [Uu: product_unit] :
                                  ( produc2624314226134418078_VEBTi
                                  @ ^ [Info3: option4927543243414619207at_nat] :
                                      ( produc7864503674192730076_VEBTi
                                      @ ^ [Deg3: nat] :
                                          ( produc9050507437146595227_VEBTi
                                          @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                              ( heap_T7982501707604696571_VEBTi
                                              @ ( refine_Imp_assert
                                                @ ( ( Info2 = Info3 )
                                                  & ( Deg2 = Deg3 ) ) )
                                              @ ^ [Uv: product_unit] :
                                                  ( produc737604151543542771_VEBTi
                                                  @ ^ [Mi4: nat,Ma4: nat] :
                                                      ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                                                      @ ^ [Mi3: nat] :
                                                          ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                                          @ ^ [Ma3: nat] :
                                                              ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ B2 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ B2 ) )
                                                              @ ^ [Xn2: nat] :
                                                                  ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ B2 @ Mi3 ) @ ( heap_Time_return_nat @ B2 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                                                  @ ^ [Minn: nat] :
                                                                      ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                      @ ^ [L: nat] :
                                                                          ( heap_T7982501707604696571_VEBTi
                                                                          @ ( refine_Imp_assert
                                                                            @ ( L
                                                                              = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ B2 @ Mi4 ) @ Mi4 @ B2 ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                          @ ^ [Uw: product_unit] :
                                                                              ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                              @ ^ [H: nat] :
                                                                                  ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                                  @ ^ [Len: nat] :
                                                                                      ( if_Hea8453224502484754311_VEBTi
                                                                                      @ ( ( ord_less_nat @ H @ Len )
                                                                                        & ~ ( ( B2 = Mi3 )
                                                                                            | ( B2 = Ma3 ) ) )
                                                                                      @ ( heap_T7982501707604696571_VEBTi
                                                                                        @ ( refine_Imp_assert
                                                                                          @ ( H
                                                                                            = ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ B2 @ Mi4 ) @ Mi4 @ B2 ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                        @ ^ [Ux: product_unit] :
                                                                                            ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                                            @ ^ [Uy: product_unit] :
                                                                                                ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                                @ ^ [Node: vEBT_VEBTi] :
                                                                                                    ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                    @ ^ [Empt: $o] :
                                                                                                        ( heap_T7982501707604696571_VEBTi
                                                                                                        @ ( refine_Imp_assert
                                                                                                          @ ( Empt
                                                                                                            = ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                                        @ ^ [Uz: product_unit] :
                                                                                                            ( heap_T1006145433769338483_VEBTi @ ( F5 @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Node @ L )
                                                                                                            @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                                ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                                                                @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                    ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( F5 @ Summary3 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                                                                    @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                        ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                                                                        @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                      @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                                                  @ ( the_Pr8591224930841456533at_nat @ Info3 ) ) ) ) )
                                  @ ( vEBT_c634343235235684882T_VEBT
                                    @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                    @ ^ [C3: $o,D3: $o] : undefi7074909574607331924T_VEBT
                                    @ X8 ) ) ) )
                        @ Info2 )
                    @ ^ [C3: $o,D3: $o] : ( if_Hea8453224502484754311_VEBTi @ ( B2 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ D3 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( B2 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ C3 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ C3 @ D3 ) ) ) )
                    @ A2 ) ) )
         => ( P @ vEBT_V3964819847710782039nserti ) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
thf(fact_7840_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_Time_Heap_nat,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A: $o,B: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A @ B ) )
         => ( time_TBOUND_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info @ Deg @ TreeArray @ Summary ) )
           => ( time_TBOUND_nat @ ( F3 @ Info @ Deg @ TreeArray @ Summary ) @ ( Bnd2 @ Info @ Deg @ TreeArray @ Summary ) ) )
       => ( time_TBOUND_nat @ ( vEBT_c1335663792808957512ap_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7841_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T2636463487746394924on_nat,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A: $o,B: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A @ B ) )
         => ( time_T8353473612707095248on_nat @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info @ Deg @ TreeArray @ Summary ) )
           => ( time_T8353473612707095248on_nat @ ( F3 @ Info @ Deg @ TreeArray @ Summary ) @ ( Bnd2 @ Info @ Deg @ TreeArray @ Summary ) ) )
       => ( time_T8353473612707095248on_nat @ ( vEBT_c6250501799366334488on_nat @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7842_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_Time_Heap_o,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A: $o,B: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A @ B ) )
         => ( time_TBOUND_o @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info @ Deg @ TreeArray @ Summary ) )
           => ( time_TBOUND_o @ ( F3 @ Info @ Deg @ TreeArray @ Summary ) @ ( Bnd2 @ Info @ Deg @ TreeArray @ Summary ) ) )
       => ( time_TBOUND_o @ ( vEBT_c6104975476656191286Heap_o @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7843_TBOUND__VEBT__case,axiom,
    ! [Ti: vEBT_VEBTi,F: $o > $o > heap_T8145700208782473153_VEBTi,Bnd: $o > $o > nat,F3: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,Bnd2: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > nat] :
      ( ! [A: $o,B: $o] :
          ( ( Ti
            = ( vEBT_Leafi @ A @ B ) )
         => ( time_T5737551269749752165_VEBTi @ ( F @ A @ B ) @ ( Bnd @ A @ B ) ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg: nat,TreeArray: array_VEBT_VEBTi,Summary: vEBT_VEBTi] :
            ( ( Ti
              = ( vEBT_Nodei @ Info @ Deg @ TreeArray @ Summary ) )
           => ( time_T5737551269749752165_VEBTi @ ( F3 @ Info @ Deg @ TreeArray @ Summary ) @ ( Bnd2 @ Info @ Deg @ TreeArray @ Summary ) ) )
       => ( time_T5737551269749752165_VEBTi @ ( vEBT_c6028912655521741485_VEBTi @ F3 @ F @ Ti ) @ ( vEBT_case_VEBTi_nat @ Bnd2 @ Bnd @ Ti ) ) ) ) ).

% TBOUND_VEBT_case
thf(fact_7844_tan__pi,axiom,
    ( ( tan_real @ pi )
    = zero_zero_real ) ).

% tan_pi
thf(fact_7845_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_7846_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_7847_tan__npi,axiom,
    ! [N2: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_7848_tan__periodic,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X2 ) ) ).

% tan_periodic
thf(fact_7849_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F2: $o > $o > heap_T2636463487746394924on_nat,X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6250501799366334488on_nat @ F1 @ F2 @ ( vEBT_Nodei @ X11 @ X122 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_7850_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F2: $o > $o > heap_Time_Heap_o,X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6104975476656191286Heap_o @ F1 @ F2 @ ( vEBT_Nodei @ X11 @ X122 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_7851_VEBTi_Osimps_I5_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_c6028912655521741485_VEBTi @ F1 @ F2 @ ( vEBT_Nodei @ X11 @ X122 @ X13 @ X14 ) )
      = ( F1 @ X11 @ X122 @ X13 @ X14 ) ) ).

% VEBTi.simps(5)
thf(fact_7852_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size(4)
thf(fact_7853_vebt__assn__raw_Ocases,axiom,
    ! [X2: produc3625547720036274456_VEBTi] :
      ( ! [A: $o,B: $o,Ai: $o,Bi: $o] :
          ( X2
         != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A @ B ) @ ( vEBT_Leafi @ Ai @ Bi ) ) )
     => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT,Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
            ( X2
           != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) )
       => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] :
              ( X2
             != ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) )
         => ~ ! [Vd3: $o,Ve3: $o,V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                ( X2
               != ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ).

% vebt_assn_raw.cases
thf(fact_7854_VEBT__internal_OminNulli_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ( X2
       != ( vEBT_Leafi @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X2
           != ( vEBT_Leafi @ $true @ Uv2 ) )
       => ( ! [Uu3: $o] :
              ( X2
             != ( vEBT_Leafi @ Uu3 @ $true ) )
         => ( ! [Uw2: nat,Ux3: array_VEBT_VEBTi,Uy3: vEBT_VEBTi] :
                ( X2
               != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
           => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                  ( X2
                 != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNulli.cases
thf(fact_7855_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T2636463487746394924on_nat,F2: $o > $o > heap_T2636463487746394924on_nat,X21: $o,X222: $o] :
      ( ( vEBT_c6250501799366334488on_nat @ F1 @ F2 @ ( vEBT_Leafi @ X21 @ X222 ) )
      = ( F2 @ X21 @ X222 ) ) ).

% VEBTi.simps(6)
thf(fact_7856_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_Time_Heap_o,F2: $o > $o > heap_Time_Heap_o,X21: $o,X222: $o] :
      ( ( vEBT_c6104975476656191286Heap_o @ F1 @ F2 @ ( vEBT_Leafi @ X21 @ X222 ) )
      = ( F2 @ X21 @ X222 ) ) ).

% VEBTi.simps(6)
thf(fact_7857_VEBTi_Osimps_I6_J,axiom,
    ! [F1: option4927543243414619207at_nat > nat > array_VEBT_VEBTi > vEBT_VEBTi > heap_T8145700208782473153_VEBTi,F2: $o > $o > heap_T8145700208782473153_VEBTi,X21: $o,X222: $o] :
      ( ( vEBT_c6028912655521741485_VEBTi @ F1 @ F2 @ ( vEBT_Leafi @ X21 @ X222 ) )
      = ( F2 @ X21 @ X222 ) ) ).

% VEBTi.simps(6)
thf(fact_7858_vebt__minti_Ocases,axiom,
    ! [X2: vEBT_VEBTi] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( vEBT_Leafi @ A @ B ) )
     => ( ! [Uu3: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
            ( X2
           != ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: array_VEBT_VEBTi,Uz3: vEBT_VEBTi] :
              ( X2
             != ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ).

% vebt_minti.cases
thf(fact_7859_TBOUND__upd,axiom,
    ! [Xs2: nat,I: vEBT_VEBTi,X2: array_VEBT_VEBTi] : ( time_T6070283812100419266_VEBTi @ ( array_upd_VEBT_VEBTi @ Xs2 @ I @ X2 ) @ one_one_nat ) ).

% TBOUND_upd
thf(fact_7860_VEBT__internal_OminNulli_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X2 )
        = Y2 )
     => ( ( ( X2
            = ( vEBT_Leafi @ $false @ $false ) )
         => ( Y2
           != ( heap_Time_return_o @ $true ) ) )
       => ( ( ? [Uv2: $o] :
                ( X2
                = ( vEBT_Leafi @ $true @ Uv2 ) )
           => ( Y2
             != ( heap_Time_return_o @ $false ) ) )
         => ( ( ? [Uu3: $o] :
                  ( X2
                  = ( vEBT_Leafi @ Uu3 @ $true ) )
             => ( Y2
               != ( heap_Time_return_o @ $false ) ) )
           => ( ( ? [Uw2: nat,Ux3: array_VEBT_VEBTi,Uy3: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
               => ( Y2
                 != ( heap_Time_return_o @ $true ) ) )
             => ~ ( ? [Uz3: product_prod_nat_nat,Va4: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( X2
                      = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                 => ( Y2
                   != ( heap_Time_return_o @ $false ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
thf(fact_7861_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(1)
thf(fact_7862_vebt__buildupi_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildupi @ zero_zero_nat )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(1)
thf(fact_7863_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    ( ( vEBT_V739175172307565963ildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(2)
thf(fact_7864_vebt__buildupi_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildupi @ ( suc @ zero_zero_nat ) )
    = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) ).

% vebt_buildupi.simps(2)
thf(fact_7865_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz2: product_prod_nat_nat,Va2: nat,Vb: array_VEBT_VEBTi,Vc: vEBT_VEBTi] :
      ( ( vEBT_VEBT_minNulli @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb @ Vc ) )
      = ( heap_Time_return_o @ $false ) ) ).

% VEBT_internal.minNulli.simps(5)
thf(fact_7866_tan__def,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).

% tan_def
thf(fact_7867_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: array_VEBT_VEBTi,Uw3: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_maxti.simps(2)
thf(fact_7868_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv3: array_VEBT_VEBTi,Uw3: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu2 @ Uv3 @ Uw3 ) )
      = ( heap_T3487192422709364219on_nat @ none_nat ) ) ).

% vebt_minti.simps(2)
thf(fact_7869_vebt__maxti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A @ B ) )
           => ~ ( ( B
                 => ( Y2
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                & ( ~ B
                 => ( ( A
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                    & ( ~ A
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux3: nat,Uy3: array_VEBT_VEBTi,Uz3: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
               => ( Y2
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) ) ) ) ) ) ).

% vebt_maxti.elims
thf(fact_7870_vebt__minti_Oelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leafi @ A @ B ) )
           => ~ ( ( A
                 => ( Y2
                    = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                & ( ~ A
                 => ( ( B
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                    & ( ~ B
                     => ( Y2
                        = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) )
       => ( ( ? [Uu3: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( X2
                = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
           => ( Y2
             != ( heap_T3487192422709364219on_nat @ none_nat ) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux3: nat,Uy3: array_VEBT_VEBTi,Uz3: vEBT_VEBTi] :
                    ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
               => ( Y2
                 != ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) ) ) ) ) ) ).

% vebt_minti.elims
thf(fact_7871_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_maxti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma ) ) ) ).

% vebt_maxti.simps(3)
thf(fact_7872_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array_VEBT_VEBTi,Uz2: vEBT_VEBTi] :
      ( ( vEBT_vebt_minti @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz2 ) )
      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi ) ) ) ).

% vebt_minti.simps(3)
thf(fact_7873_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7874_vebt__maxti_Osimps_I1_J,axiom,
    ! [B3: $o,A3: $o] :
      ( ( B3
       => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
      & ( ~ B3
       => ( ( A3
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
          & ( ~ A3
           => ( ( vEBT_vebt_maxti @ ( vEBT_Leafi @ A3 @ B3 ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_maxti.simps(1)
thf(fact_7875_vebt__minti_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( A3
       => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
      & ( ~ A3
       => ( ( B3
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
              = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
          & ( ~ B3
           => ( ( vEBT_vebt_minti @ ( vEBT_Leafi @ A3 @ B3 ) )
              = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) ) ).

% vebt_minti.simps(1)
thf(fact_7876_lemma__tan__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y2 @ ( tan_real @ X3 ) ) ) ) ).

% lemma_tan_total
thf(fact_7877_tan__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_gt_zero
thf(fact_7878_lemma__tan__total1,axiom,
    ! [Y2: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y2 ) ) ).

% lemma_tan_total1
thf(fact_7879_tan__mono__lt__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
              = ( ord_less_real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_7880_tan__monotone_H,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
         => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y2 @ X2 )
              = ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_7881_tan__monotone,axiom,
    ! [Y2: real,X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ X2 )
       => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y2 ) @ ( tan_real @ X2 ) ) ) ) ) ).

% tan_monotone
thf(fact_7882_tan__total,axiom,
    ! [Y2: real] :
    ? [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X3 )
        = Y2 )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y2 ) )
         => ( Y4 = X3 ) ) ) ).

% tan_total
thf(fact_7883_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_7884_tan__inverse,axiom,
    ! [Y2: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y2 ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 ) ) ) ).

% tan_inverse
thf(fact_7885_vebt__inserti_Omono,axiom,
    ! [X2: produc3881548065746020326Ti_nat] :
      ( comple2284608890766496472_VEBTi @ ( partia6690842624828592406Ti_nat @ heap_T7173139186834293313_VEBTi ) @ heap_T7173139186834293313_VEBTi
      @ ^ [Vebt_inserti2: produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi] :
          ( produc3255295512018472142_VEBTi
          @ ^ [T: vEBT_VEBTi,X: nat] :
              ( vEBT_c6028912655521741485_VEBTi
              @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                  ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                  @ ^ [Minma: product_prod_nat_nat] :
                      ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                      @ ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                        @ ^ [Mi3: nat] :
                            ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                            @ ^ [Ma3: nat] :
                                ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X ) )
                                @ ^ [Xn2: nat] :
                                    ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ X ) @ ( heap_Time_return_nat @ Mi3 ) )
                                    @ ^ [Minn: nat] :
                                        ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [L: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [H: nat] :
                                                ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                @ ^ [Len: nat] :
                                                    ( if_Hea8453224502484754311_VEBTi
                                                    @ ( ( ord_less_nat @ H @ Len )
                                                      & ~ ( ( X = Mi3 )
                                                          | ( X = Ma3 ) ) )
                                                    @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                      @ ^ [Node: vEBT_VEBTi] :
                                                          ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                          @ ^ [Empt: $o] :
                                                              ( heap_T1006145433769338483_VEBTi @ ( produc5159149307777246319_VEBTi @ Vebt_inserti2 @ Node @ L )
                                                              @ ^ [Newnode2: vEBT_VEBTi] :
                                                                  ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                  @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                      ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( produc5159149307777246319_VEBTi @ Vebt_inserti2 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                      @ ^ [Newsummary: vEBT_VEBTi] :
                                                                          ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                          @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                    @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                  @ Info2 )
              @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
              @ T )
          @ X2 ) ) ).

% vebt_inserti.mono
thf(fact_7886_add__tan__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7887_exE__realizer_H,axiom,
    ! [P: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( P @ ( product_snd_nat_nat @ P2 ) @ ( product_fst_nat_nat @ P2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7888_exE__realizer_H,axiom,
    ! [P: int > int > $o,P2: product_prod_int_int] :
      ( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
     => ~ ! [X3: int,Y3: int] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7889_exE__realizer_H,axiom,
    ! [P: assn > assn > $o,P2: produc6575502325842934193n_assn] :
      ( ( P @ ( produc2051961928117032727n_assn @ P2 ) @ ( produc9167289414957590229n_assn @ P2 ) )
     => ~ ! [X3: assn,Y3: assn] :
            ~ ( P @ Y3 @ X3 ) ) ).

% exE_realizer'
thf(fact_7890_vebt__inserti_Oraw__induct,axiom,
    ! [P: vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o,Xa: produc3881548065746020326Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti3: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,N5: nat] :
              ( ( heap_T2071195472996403633_VEBTi @ ( Vebt_inserti3 @ A6 @ B5 ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Xa2: heap_e7401611519738050253t_unit,R3: vEBT_VEBTi,N4: nat] :
              ( ( heap_T2071195472996403633_VEBTi
                @ ( vEBT_c6028912655521741485_VEBTi
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                      @ ^ [Minma: product_prod_nat_nat] :
                          ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                          @ ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                            @ ^ [Mi3: nat] :
                                ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                @ ^ [Ma3: nat] :
                                    ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X3 ) )
                                    @ ^ [Xn2: nat] :
                                        ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_nat @ X3 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                        @ ^ [Minn: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                            @ ^ [L: nat] :
                                                ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                @ ^ [H: nat] :
                                                    ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                    @ ^ [Len: nat] :
                                                        ( if_Hea8453224502484754311_VEBTi
                                                        @ ( ( ord_less_nat @ H @ Len )
                                                          & ~ ( ( X3 = Mi3 )
                                                              | ( X3 = Ma3 ) ) )
                                                        @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                          @ ^ [Node: vEBT_VEBTi] :
                                                              ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                              @ ^ [Empt: $o] :
                                                                  ( heap_T1006145433769338483_VEBTi @ ( Vebt_inserti3 @ Node @ L )
                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                      ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                      @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                          ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( Vebt_inserti3 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                          @ ^ [Newsummary: vEBT_VEBTi] :
                                                                              ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                              @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                        @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                      @ Info2 )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X3 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X3 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
                  @ T3 )
                @ Ta
                @ Xa2
                @ R3
                @ N4 )
             => ( P @ T3 @ X3 @ Ta @ Xa2 @ R3 @ N4 ) ) )
     => ( ( heap_T2071195472996403633_VEBTi @ ( produc3255295512018472142_VEBTi @ vEBT_vebt_inserti @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc4924893227731358948_nat_o @ P @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% vebt_inserti.raw_induct
thf(fact_7891_vebt__inserti_Osimps,axiom,
    ( vEBT_vebt_inserti
    = ( ^ [T: vEBT_VEBTi,X: nat] :
          ( vEBT_c6028912655521741485_VEBTi
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
              @ ^ [Minma: product_prod_nat_nat] :
                  ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                  @ ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                    @ ^ [Mi3: nat] :
                        ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                        @ ^ [Ma3: nat] :
                            ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X ) )
                            @ ^ [Xn2: nat] :
                                ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ X ) @ ( heap_Time_return_nat @ Mi3 ) )
                                @ ^ [Minn: nat] :
                                    ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [L: nat] :
                                        ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [H: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                            @ ^ [Len: nat] :
                                                ( if_Hea8453224502484754311_VEBTi
                                                @ ( ( ord_less_nat @ H @ Len )
                                                  & ~ ( ( X = Mi3 )
                                                      | ( X = Ma3 ) ) )
                                                @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                  @ ^ [Node: vEBT_VEBTi] :
                                                      ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                      @ ^ [Empt: $o] :
                                                          ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_inserti @ Node @ L )
                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                              ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                              @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                  ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( vEBT_vebt_inserti @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                  @ ^ [Newsummary: vEBT_VEBTi] :
                                                                      ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                      @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
              @ Info2 )
          @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
          @ T ) ) ) ).

% vebt_inserti.simps
thf(fact_7892_vebt__inserti_Ofixp__induct,axiom,
    ! [P: ( vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi ) > $o] :
      ( ( comple380401974140132787_VEBTi @ ( partia6972460264168101086Ti_nat @ heap_T3112222404744780921_VEBTi ) @ ( partia6690842624828592406Ti_nat @ heap_T7173139186834293313_VEBTi )
        @ ^ [Vebt_inserti2: produc3881548065746020326Ti_nat > heap_T8145700208782473153_VEBTi] : ( P @ ( produc5159149307777246319_VEBTi @ Vebt_inserti2 ) ) )
     => ( ( P
          @ ^ [Vebt_inserti2: vEBT_VEBTi,T: nat] :
              ( heap_T1489671670754571048_VEBTi
              @ ^ [X: heap_e7401611519738050253t_unit] : none_P7832717587476222275it_nat ) )
       => ( ! [F5: vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
              ( ( P @ F5 )
             => ( P
                @ ^ [X8: vEBT_VEBTi,A2: nat] :
                    ( vEBT_c6028912655521741485_VEBTi
                    @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                        ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                        @ ^ [Minma: product_prod_nat_nat] :
                            ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                            @ ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                              @ ^ [Mi3: nat] :
                                  ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                  @ ^ [Ma3: nat] :
                                      ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ A2 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ A2 ) )
                                      @ ^ [Xn2: nat] :
                                          ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ A2 @ Mi3 ) @ ( heap_Time_return_nat @ A2 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                          @ ^ [Minn: nat] :
                                              ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                              @ ^ [L: nat] :
                                                  ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                  @ ^ [H: nat] :
                                                      ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                      @ ^ [Len: nat] :
                                                          ( if_Hea8453224502484754311_VEBTi
                                                          @ ( ( ord_less_nat @ H @ Len )
                                                            & ~ ( ( A2 = Mi3 )
                                                                | ( A2 = Ma3 ) ) )
                                                          @ ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                            @ ^ [Node: vEBT_VEBTi] :
                                                                ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                @ ^ [Empt: $o] :
                                                                    ( heap_T1006145433769338483_VEBTi @ ( F5 @ Node @ L )
                                                                    @ ^ [Newnode2: vEBT_VEBTi] :
                                                                        ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                        @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                            ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( F5 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                            @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                                @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) )
                                                          @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                        @ Info2 )
                    @ ^ [B2: $o,C3: $o] : ( if_Hea8453224502484754311_VEBTi @ ( A2 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ C3 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( A2 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ B2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ B2 @ C3 ) ) ) )
                    @ X8 ) ) )
         => ( P @ vEBT_vebt_inserti ) ) ) ) ).

% vebt_inserti.fixp_induct
thf(fact_7893_tan__pos__pi2__le,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_7894_tan__total__pos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X3 )
            = Y2 ) ) ) ).

% tan_total_pos
thf(fact_7895_tan__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_7896_tan__mono__le,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ).

% tan_mono_le
thf(fact_7897_tan__mono__le__eq,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y2 )
         => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) )
              = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_7898_tan__bound__pi2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_7899_arctan,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y2 ) )
      & ( ord_less_real @ ( arctan @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y2 ) )
        = Y2 ) ) ).

% arctan
thf(fact_7900_arctan__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X2 ) )
          = X2 ) ) ) ).

% arctan_tan
thf(fact_7901_arctan__unique,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X2 )
            = Y2 )
         => ( ( arctan @ Y2 )
            = X2 ) ) ) ) ).

% arctan_unique
thf(fact_7902_lemma__tan__add1,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y2 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7903_tan__diff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X2 @ Y2 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X2 @ Y2 ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7904_tan__add,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( cos_real @ X2 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y2 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X2 @ Y2 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X2 @ Y2 ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X2 ) @ ( tan_real @ Y2 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7905_tan__total__pi4,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ? [Z4: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
          & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z4 )
            = X2 ) ) ) ).

% tan_total_pi4
thf(fact_7906_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R4: option_num > $o,F: option_num,G: num > option_num] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R4 @ ( G @ Q3 ) ) )
         => ( R4 @ ( case_o6005452278849405969um_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7907_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R4: num > $o,F: num,G: num > num] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R4 @ ( G @ Q3 ) ) )
         => ( R4 @ ( case_option_num_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7908_disjE__realizer2,axiom,
    ! [P: $o,Q: num > $o,X2: option_num,R4: int > $o,F: int,G: num > int] :
      ( ( case_option_o_num @ P @ Q @ X2 )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q3: num] :
              ( ( Q @ Q3 )
             => ( R4 @ ( G @ Q3 ) ) )
         => ( R4 @ ( case_option_int_num @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7909_disjE__realizer2,axiom,
    ! [P: $o,Q: product_prod_nat_nat > $o,X2: option4927543243414619207at_nat,R4: heap_Time_Heap_o > $o,F: heap_Time_Heap_o,G: product_prod_nat_nat > heap_Time_Heap_o] :
      ( ( case_o184042715313410164at_nat @ P @ Q @ X2 )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( ( Q @ Q3 )
             => ( R4 @ ( G @ Q3 ) ) )
         => ( R4 @ ( case_o1442776274061689234at_nat @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7910_disjE__realizer2,axiom,
    ! [P: $o,Q: product_prod_nat_nat > $o,X2: option4927543243414619207at_nat,R4: $o > $o,F: $o,G: product_prod_nat_nat > $o] :
      ( ( case_o184042715313410164at_nat @ P @ Q @ X2 )
     => ( ( P
         => ( R4 @ F ) )
       => ( ! [Q3: product_prod_nat_nat] :
              ( ( Q @ Q3 )
             => ( R4 @ ( G @ Q3 ) ) )
         => ( R4 @ ( case_o184042715313410164at_nat @ F @ G @ X2 ) ) ) ) ) ).

% disjE_realizer2
thf(fact_7911_tan__half,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7912_exI__realizer,axiom,
    ! [P: num > num > $o,Y2: num,X2: num] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( product_snd_num_num @ ( product_Pair_num_num @ X2 @ Y2 ) ) @ ( product_fst_num_num @ ( product_Pair_num_num @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7913_exI__realizer,axiom,
    ! [P: produc4813437837504472865T_VEBT > nat > $o,Y2: produc4813437837504472865T_VEBT,X2: nat] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc2084898568784432842T_VEBT @ ( produc1750349459881913976T_VEBT @ X2 @ Y2 ) ) @ ( produc758997459209783180T_VEBT @ ( produc1750349459881913976T_VEBT @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7914_exI__realizer,axiom,
    ! [P: num > nat > $o,Y2: num,X2: nat] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( product_snd_nat_num @ ( product_Pair_nat_num @ X2 @ Y2 ) ) @ ( product_fst_nat_num @ ( product_Pair_nat_num @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7915_exI__realizer,axiom,
    ! [P: nat > nat > $o,Y2: nat,X2: nat] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7916_exI__realizer,axiom,
    ! [P: int > int > $o,Y2: int,X2: int] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7917_exI__realizer,axiom,
    ! [P: assn > assn > $o,Y2: assn,X2: assn] :
      ( ( P @ Y2 @ X2 )
     => ( P @ ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) ) @ ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ X2 @ Y2 ) ) ) ) ).

% exI_realizer
thf(fact_7918_conjI__realizer,axiom,
    ! [P: num > $o,P2: num,Q: num > $o,Q2: num] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_num_num @ ( product_Pair_num_num @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_num_num @ ( product_Pair_num_num @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7919_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: produc4813437837504472865T_VEBT > $o,Q2: produc4813437837504472865T_VEBT] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc758997459209783180T_VEBT @ ( produc1750349459881913976T_VEBT @ P2 @ Q2 ) ) )
          & ( Q @ ( produc2084898568784432842T_VEBT @ ( produc1750349459881913976T_VEBT @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7920_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: num > $o,Q2: num] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_num @ ( product_Pair_nat_num @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_num @ ( product_Pair_nat_num @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7921_conjI__realizer,axiom,
    ! [P: nat > $o,P2: nat,Q: nat > $o,Q2: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_nat_nat @ ( product_Pair_nat_nat @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7922_conjI__realizer,axiom,
    ! [P: int > $o,P2: int,Q: int > $o,Q2: int] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P2 @ Q2 ) ) )
          & ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7923_conjI__realizer,axiom,
    ! [P: assn > $o,P2: assn,Q: assn > $o,Q2: assn] :
      ( ( P @ P2 )
     => ( ( Q @ Q2 )
       => ( ( P @ ( produc9167289414957590229n_assn @ ( produc118845697133431529n_assn @ P2 @ Q2 ) ) )
          & ( Q @ ( produc2051961928117032727n_assn @ ( produc118845697133431529n_assn @ P2 @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_7924_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [X2: produc3960855945107176009Ti_nat] :
      ( comple5606513277678308283_VEBTi @ ( partia1868168049876374393Ti_nat @ heap_T7173139186834293313_VEBTi ) @ heap_T7173139186834293313_VEBTi
      @ ^ [Vebt_inserti: produc3960855945107176009Ti_nat > heap_T8145700208782473153_VEBTi] :
          ( produc2943724498215716011_VEBTi
          @ ( produc2298712477539903273_VEBTi
            @ ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
                ( vEBT_c6028912655521741485_VEBTi
                @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                    ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                    @ ^ [Minma: product_prod_nat_nat] :
                        ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                        @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
                          @ ^ [Uu: product_unit] :
                              ( produc2624314226134418078_VEBTi
                              @ ^ [Info3: option4927543243414619207at_nat] :
                                  ( produc7864503674192730076_VEBTi
                                  @ ^ [Deg3: nat] :
                                      ( produc9050507437146595227_VEBTi
                                      @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                          ( heap_T7982501707604696571_VEBTi
                                          @ ( refine_Imp_assert
                                            @ ( ( Info2 = Info3 )
                                              & ( Deg2 = Deg3 ) ) )
                                          @ ^ [Uv: product_unit] :
                                              ( produc737604151543542771_VEBTi
                                              @ ^ [Mi4: nat,Ma4: nat] :
                                                  ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                                                  @ ^ [Mi3: nat] :
                                                      ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                                      @ ^ [Ma3: nat] :
                                                          ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X ) )
                                                          @ ^ [Xn2: nat] :
                                                              ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ X ) @ ( heap_Time_return_nat @ Mi3 ) )
                                                              @ ^ [Minn: nat] :
                                                                  ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                  @ ^ [L: nat] :
                                                                      ( heap_T7982501707604696571_VEBTi
                                                                      @ ( refine_Imp_assert
                                                                        @ ( L
                                                                          = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                      @ ^ [Uw: product_unit] :
                                                                          ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                          @ ^ [H: nat] :
                                                                              ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                              @ ^ [Len: nat] :
                                                                                  ( if_Hea8453224502484754311_VEBTi
                                                                                  @ ( ( ord_less_nat @ H @ Len )
                                                                                    & ~ ( ( X = Mi3 )
                                                                                        | ( X = Ma3 ) ) )
                                                                                  @ ( heap_T7982501707604696571_VEBTi
                                                                                    @ ( refine_Imp_assert
                                                                                      @ ( H
                                                                                        = ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                    @ ^ [Ux: product_unit] :
                                                                                        ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                                        @ ^ [Uy: product_unit] :
                                                                                            ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                            @ ^ [Node: vEBT_VEBTi] :
                                                                                                ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                @ ^ [Empt: $o] :
                                                                                                    ( heap_T7982501707604696571_VEBTi
                                                                                                    @ ( refine_Imp_assert
                                                                                                      @ ( Empt
                                                                                                        = ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                                    @ ^ [Uz: product_unit] :
                                                                                                        ( heap_T1006145433769338483_VEBTi @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti ) @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Node @ L )
                                                                                                        @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                            ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                                                            @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( produc4062687810467144008_VEBTi @ ( produc2164094337957399884_VEBTi @ Vebt_inserti ) @ Summary3 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                                                                @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                    ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                                                                    @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                  @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                                              @ ( the_Pr8591224930841456533at_nat @ Info3 ) ) ) ) )
                              @ ( vEBT_c634343235235684882T_VEBT
                                @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                                @ T ) ) ) )
                    @ Info2 )
                @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
                @ Ti3 ) )
          @ X2 ) ) ).

% VEBT_internal.vebt_inserti'.mono
thf(fact_7925_VEBT__internal_Ovebt__inserti_H_Oraw__induct,axiom,
    ! [P: vEBT_VEBT > vEBT_VEBTi > nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > vEBT_VEBTi > nat > $o,Xa: produc3960855945107176009Ti_nat,H2: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,R: vEBT_VEBTi,N2: nat] :
      ( ! [Vebt_inserti4: vEBT_VEBT > vEBT_VEBTi > nat > heap_T8145700208782473153_VEBTi] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,R2: vEBT_VEBTi,N5: nat] :
              ( ( heap_T2071195472996403633_VEBTi @ ( Vebt_inserti4 @ A6 @ B5 @ Ba ) @ H4 @ H5 @ R2 @ N5 )
             => ( P @ A6 @ B5 @ Ba @ H4 @ H5 @ R2 @ N5 ) )
         => ! [T3: vEBT_VEBT,Ti4: vEBT_VEBTi,X3: nat,Ta: heap_e7401611519738050253t_unit,Tia: heap_e7401611519738050253t_unit,Xa2: vEBT_VEBTi,N4: nat] :
              ( ( heap_T2071195472996403633_VEBTi
                @ ( vEBT_c6028912655521741485_VEBTi
                  @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
                      ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
                      @ ^ [Minma: product_prod_nat_nat] :
                          ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                          @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T3 ) )
                            @ ^ [Uu: product_unit] :
                                ( produc2624314226134418078_VEBTi
                                @ ^ [Info3: option4927543243414619207at_nat] :
                                    ( produc7864503674192730076_VEBTi
                                    @ ^ [Deg3: nat] :
                                        ( produc9050507437146595227_VEBTi
                                        @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                            ( heap_T7982501707604696571_VEBTi
                                            @ ( refine_Imp_assert
                                              @ ( ( Info2 = Info3 )
                                                & ( Deg2 = Deg3 ) ) )
                                            @ ^ [Uv: product_unit] :
                                                ( produc737604151543542771_VEBTi
                                                @ ^ [Mi4: nat,Ma4: nat] :
                                                    ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                                                    @ ^ [Mi3: nat] :
                                                        ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                                        @ ^ [Ma3: nat] :
                                                            ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X3 ) )
                                                            @ ^ [Xn2: nat] :
                                                                ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X3 @ Mi3 ) @ ( heap_Time_return_nat @ X3 ) @ ( heap_Time_return_nat @ Mi3 ) )
                                                                @ ^ [Minn: nat] :
                                                                    ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                    @ ^ [L: nat] :
                                                                        ( heap_T7982501707604696571_VEBTi
                                                                        @ ( refine_Imp_assert
                                                                          @ ( L
                                                                            = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X3 @ Mi4 ) @ Mi4 @ X3 ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                        @ ^ [Uw: product_unit] :
                                                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                            @ ^ [H: nat] :
                                                                                ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                                @ ^ [Len: nat] :
                                                                                    ( if_Hea8453224502484754311_VEBTi
                                                                                    @ ( ( ord_less_nat @ H @ Len )
                                                                                      & ~ ( ( X3 = Mi3 )
                                                                                          | ( X3 = Ma3 ) ) )
                                                                                    @ ( heap_T7982501707604696571_VEBTi
                                                                                      @ ( refine_Imp_assert
                                                                                        @ ( H
                                                                                          = ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X3 @ Mi4 ) @ Mi4 @ X3 ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                                      @ ^ [Ux: product_unit] :
                                                                                          ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                                          @ ^ [Uy: product_unit] :
                                                                                              ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                              @ ^ [Node: vEBT_VEBTi] :
                                                                                                  ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                                  @ ^ [Empt: $o] :
                                                                                                      ( heap_T7982501707604696571_VEBTi
                                                                                                      @ ( refine_Imp_assert
                                                                                                        @ ( Empt
                                                                                                          = ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                                      @ ^ [Uz: product_unit] :
                                                                                                          ( heap_T1006145433769338483_VEBTi @ ( Vebt_inserti4 @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Node @ L )
                                                                                                          @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                              ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                                                              @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                                  ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( Vebt_inserti4 @ Summary3 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                                                                  @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                                      ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                                                                      @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                                    @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                                                @ ( the_Pr8591224930841456533at_nat @ Info3 ) ) ) ) )
                                @ ( vEBT_c634343235235684882T_VEBT
                                  @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                                  @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                                  @ T3 ) ) ) )
                      @ Info2 )
                  @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X3 = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X3 = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
                  @ Ti4 )
                @ Ta
                @ Tia
                @ Xa2
                @ N4 )
             => ( P @ T3 @ Ti4 @ X3 @ Ta @ Tia @ Xa2 @ N4 ) ) )
     => ( ( heap_T2071195472996403633_VEBTi @ ( produc2943724498215716011_VEBTi @ ( produc2298712477539903273_VEBTi @ vEBT_V3964819847710782039nserti ) @ Xa ) @ H2 @ H3 @ R @ N2 )
       => ( produc7403044070069621057_nat_o @ ( produc2677327216024927295_nat_o @ P ) @ Xa @ H2 @ H3 @ R @ N2 ) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
thf(fact_7926_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ( vEBT_V3964819847710782039nserti
    = ( ^ [T: vEBT_VEBT,Ti3: vEBT_VEBTi,X: nat] :
          ( vEBT_c6028912655521741485_VEBTi
          @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
              ( case_o1356590567247012107at_nat @ ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) )
              @ ^ [Minma: product_prod_nat_nat] :
                  ( if_Hea8453224502484754311_VEBTi @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ Info2 @ Deg2 @ TreeArray2 @ Summary2 ) )
                  @ ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( vEBT_is_Node @ T ) )
                    @ ^ [Uu: product_unit] :
                        ( produc2624314226134418078_VEBTi
                        @ ^ [Info3: option4927543243414619207at_nat] :
                            ( produc7864503674192730076_VEBTi
                            @ ^ [Deg3: nat] :
                                ( produc9050507437146595227_VEBTi
                                @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                                    ( heap_T7982501707604696571_VEBTi
                                    @ ( refine_Imp_assert
                                      @ ( ( Info2 = Info3 )
                                        & ( Deg2 = Deg3 ) ) )
                                    @ ^ [Uv: product_unit] :
                                        ( produc737604151543542771_VEBTi
                                        @ ^ [Mi4: nat,Ma4: nat] :
                                            ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_fst_nat_nat @ Minma ) )
                                            @ ^ [Mi3: nat] :
                                                ( heap_T844888390831797134_VEBTi @ ( heap_Time_return_nat @ ( product_snd_nat_nat @ Minma ) )
                                                @ ^ [Ma3: nat] :
                                                    ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ Mi3 ) @ ( heap_Time_return_nat @ X ) )
                                                    @ ^ [Xn2: nat] :
                                                        ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ X @ Mi3 ) @ ( heap_Time_return_nat @ X ) @ ( heap_Time_return_nat @ Mi3 ) )
                                                        @ ^ [Minn: nat] :
                                                            ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_lowi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                            @ ^ [L: nat] :
                                                                ( heap_T7982501707604696571_VEBTi
                                                                @ ( refine_Imp_assert
                                                                  @ ( L
                                                                    = ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                @ ^ [Uw: product_unit] :
                                                                    ( heap_T844888390831797134_VEBTi @ ( vEBT_VEBT_highi @ Xn2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                                                    @ ^ [H: nat] :
                                                                        ( heap_T844888390831797134_VEBTi @ ( array_len_VEBT_VEBTi @ TreeArray2 )
                                                                        @ ^ [Len: nat] :
                                                                            ( if_Hea8453224502484754311_VEBTi
                                                                            @ ( ( ord_less_nat @ H @ Len )
                                                                              & ~ ( ( X = Mi3 )
                                                                                  | ( X = Ma3 ) ) )
                                                                            @ ( heap_T7982501707604696571_VEBTi
                                                                              @ ( refine_Imp_assert
                                                                                @ ( H
                                                                                  = ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi4 ) @ Mi4 @ X ) @ ( divide_divide_nat @ Deg3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                                              @ ^ [Ux: product_unit] :
                                                                                  ( heap_T7982501707604696571_VEBTi @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                                                  @ ^ [Uy: product_unit] :
                                                                                      ( heap_T1006145433769338483_VEBTi @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                                                      @ ^ [Node: vEBT_VEBTi] :
                                                                                          ( heap_T5998771940306268294_VEBTi @ ( vEBT_VEBT_minNulli @ Node )
                                                                                          @ ^ [Empt: $o] :
                                                                                              ( heap_T7982501707604696571_VEBTi
                                                                                              @ ( refine_Imp_assert
                                                                                                @ ( Empt
                                                                                                  = ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                                              @ ^ [Uz: product_unit] :
                                                                                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_V3964819847710782039nserti @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Node @ L )
                                                                                                  @ ^ [Newnode2: vEBT_VEBTi] :
                                                                                                      ( heap_T5099337393651448672_VEBTi @ ( array_upd_VEBT_VEBTi @ H @ Newnode2 @ TreeArray2 )
                                                                                                      @ ^ [Newarray: array_VEBT_VEBTi] :
                                                                                                          ( heap_T1006145433769338483_VEBTi @ ( if_Hea8453224502484754311_VEBTi @ Empt @ ( vEBT_V3964819847710782039nserti @ Summary3 @ Summary2 @ H ) @ ( heap_T3630416162098727440_VEBTi @ Summary2 ) )
                                                                                                          @ ^ [Newsummary: vEBT_VEBTi] :
                                                                                                              ( heap_T844888390831797134_VEBTi @ ( if_Hea2662716070787841314ap_nat @ ( ord_less_nat @ Ma3 @ Xn2 ) @ ( heap_Time_return_nat @ Xn2 ) @ ( heap_Time_return_nat @ Ma3 ) )
                                                                                                              @ ^ [Man: nat] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Minn @ Man ) ) @ Deg2 @ Newarray @ Newsummary ) ) ) ) ) ) ) ) ) ) )
                                                                            @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeArray2 @ Summary2 ) ) ) ) ) ) ) ) ) ) )
                                        @ ( the_Pr8591224930841456533at_nat @ Info3 ) ) ) ) )
                        @ ( vEBT_c634343235235684882T_VEBT
                          @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                          @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                          @ T ) ) ) )
              @ Info2 )
          @ ^ [A2: $o,B2: $o] : ( if_Hea8453224502484754311_VEBTi @ ( X = zero_zero_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $true @ B2 ) ) @ ( if_Hea8453224502484754311_VEBTi @ ( X = one_one_nat ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ $true ) ) @ ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ A2 @ B2 ) ) ) )
          @ Ti3 ) ) ) ).

% VEBT_internal.vebt_inserti'.simps
thf(fact_7927_VEBT__internal_Ovebt__buildupi_H_Oelims,axiom,
    ! [X2: nat,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( ( vEBT_V739175172307565963ildupi @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ^ [TreeList: list_VEBT_VEBTi] :
                              ( heap_T7982501707604696571_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              @ ^ [Uu: product_unit] :
                                  ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                  @ ^ [Trees: array_VEBT_VEBTi] :
                                      ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ^ [TreeList: list_VEBT_VEBTi] :
                              ( heap_T7982501707604696571_VEBTi
                              @ ( refine_Imp_assert
                                @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                              @ ^ [Uu: product_unit] :
                                  ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                  @ ^ [Trees: array_VEBT_VEBTi] :
                                      ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
thf(fact_7928_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va2 ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList: list_VEBT_VEBTi] :
                ( heap_T7982501707604696571_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ^ [Uu: product_unit] :
                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                    @ ^ [Trees: array_VEBT_VEBTi] :
                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                        @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_V739175172307565963ildupi @ ( suc @ ( suc @ Va2 ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList: list_VEBT_VEBTi] :
                ( heap_T7982501707604696571_VEBTi
                @ ( refine_Imp_assert
                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                @ ^ [Uu: product_unit] :
                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                    @ ^ [Trees: array_VEBT_VEBTi] :
                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                        @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
thf(fact_7929_vebt__buildupi_Oelims,axiom,
    ! [X2: nat,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( ( vEBT_vebt_buildupi @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ^ [TreeList: list_VEBT_VEBTi] :
                              ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                              @ ^ [Trees: array_VEBT_VEBTi] :
                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                  @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ^ [TreeList: list_VEBT_VEBTi] :
                              ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                              @ ^ [Trees: array_VEBT_VEBTi] :
                                  ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                  @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
thf(fact_7930_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va2 ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList: list_VEBT_VEBTi] :
                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                @ ^ [Trees: array_VEBT_VEBTi] :
                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                    @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ Trees @ Summary2 ) ) ) ) ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildupi @ ( suc @ ( suc @ Va2 ) ) )
          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
            @ ^ [TreeList: list_VEBT_VEBTi] :
                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                @ ^ [Trees: array_VEBT_VEBTi] :
                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) ) ).

% vebt_buildupi.simps(3)
thf(fact_7931_in__set__enumerate__eq,axiom,
    ! [P2: produc8025551001238799321T_VEBT,N2: nat,Xs2: list_VEBT_VEBT] :
      ( ( member8549952807677709168T_VEBT @ P2 @ ( set_Pr5984661752051438084T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( produc8575180428842422559T_VEBT @ P2 ) )
        & ( ord_less_nat @ ( produc8575180428842422559T_VEBT @ P2 ) @ ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) )
        & ( ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ ( produc8575180428842422559T_VEBT @ P2 ) @ N2 ) )
          = ( produc8172668247895388509T_VEBT @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7932_in__set__enumerate__eq,axiom,
    ! [P2: produc214224863196444774_VEBTi,N2: nat,Xs2: list_VEBT_VEBTi] :
      ( ( member763447850065367567_VEBTi @ P2 @ ( set_Pr4207466110102731387_VEBTi @ ( enumerate_VEBT_VEBTi @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( produc8252055991070844170_VEBTi @ P2 ) )
        & ( ord_less_nat @ ( produc8252055991070844170_VEBTi @ P2 ) @ ( plus_plus_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ N2 ) )
        & ( ( nth_VEBT_VEBTi @ Xs2 @ ( minus_minus_nat @ ( produc8252055991070844170_VEBTi @ P2 ) @ N2 ) )
          = ( produc271786961351840588_VEBTi @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7933_in__set__enumerate__eq,axiom,
    ! [P2: produc7716430852924023517t_real,N2: nat,Xs2: list_real] :
      ( ( member557208447399453958t_real @ P2 @ ( set_Pr7149346036329476978t_real @ ( enumerate_real @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_real @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_real @ P2 ) @ ( plus_plus_nat @ ( size_size_list_real @ Xs2 ) @ N2 ) )
        & ( ( nth_real @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_real @ P2 ) @ N2 ) )
          = ( product_snd_nat_real @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7934_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_o,N2: nat,Xs2: list_o] :
      ( ( member6310962623043647828_nat_o @ P2 @ ( set_Pr1291962091234853352_nat_o @ ( enumerate_o @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_o @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_o @ P2 ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) )
        & ( ( nth_o @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_o @ P2 ) @ N2 ) )
          = ( product_snd_nat_o @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7935_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_int,N2: nat,Xs2: list_int] :
      ( ( member4262671552274231302at_int @ P2 @ ( set_Pr1470767568048878706at_int @ ( enumerate_int @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_int @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_int @ P2 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) )
        & ( ( nth_int @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_int @ P2 ) @ N2 ) )
          = ( product_snd_nat_int @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7936_in__set__enumerate__eq,axiom,
    ! [P2: product_prod_nat_nat,N2: nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( enumerate_nat @ N2 @ Xs2 ) ) )
      = ( ( ord_less_eq_nat @ N2 @ ( product_fst_nat_nat @ P2 ) )
        & ( ord_less_nat @ ( product_fst_nat_nat @ P2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) )
        & ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( product_fst_nat_nat @ P2 ) @ N2 ) )
          = ( product_snd_nat_nat @ P2 ) ) ) ) ).

% in_set_enumerate_eq
thf(fact_7937_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_real] :
      ( ( size_s7910714270633306959t_real @ ( enumerate_real @ N2 @ Xs2 ) )
      = ( size_size_list_real @ Xs2 ) ) ).

% length_enumerate
thf(fact_7938_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N2 @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_enumerate
thf(fact_7939_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_7940_length__enumerate,axiom,
    ! [N2: nat,Xs2: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_enumerate
thf(fact_7941_map__snd__enumerate,axiom,
    ! [N2: nat,Xs2: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( enumerate_nat @ N2 @ Xs2 ) )
      = Xs2 ) ).

% map_snd_enumerate
thf(fact_7942_TBOUND__of__list,axiom,
    ! [Xs2: list_VEBT_VEBTi] : ( time_T6070283812100419266_VEBTi @ ( array_615059503499738533_VEBTi @ Xs2 ) @ ( suc @ ( size_s7982070591426661849_VEBTi @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_7943_TBOUND__of__list,axiom,
    ! [Xs2: list_o] : ( time_TBOUND_array_o @ ( array_of_list_o @ Xs2 ) @ ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_7944_TBOUND__of__list,axiom,
    ! [Xs2: list_nat] : ( time_T3621710982123377501ay_nat @ ( array_of_list_nat @ Xs2 ) @ ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_7945_TBOUND__of__list,axiom,
    ! [Xs2: list_int] : ( time_T8667231999468956601ay_int @ ( array_of_list_int @ Xs2 ) @ ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% TBOUND_of_list
thf(fact_7946_time__array__of__list,axiom,
    ! [Xs2: list_VEBT_VEBTi,H2: heap_e7401611519738050253t_unit] :
      ( ( time_t9122064381910598399_VEBTi @ ( array_615059503499738533_VEBTi @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_s7982070591426661849_VEBTi @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_7947_time__array__of__list,axiom,
    ! [Xs2: list_o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_o @ ( array_of_list_o @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_7948_time__array__of__list,axiom,
    ! [Xs2: list_nat,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_nat @ ( array_of_list_nat @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_7949_time__array__of__list,axiom,
    ! [Xs2: list_int,H2: heap_e7401611519738050253t_unit] :
      ( ( time_time_array_int @ ( array_of_list_int @ Xs2 ) @ H2 )
      = ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat ) ) ).

% time_array_of_list
thf(fact_7950_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBT,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs2 ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBT @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7951_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_VEBT_VEBTi,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_s7982070591426661849_VEBTi @ Xs2 ) )
     => ( ( nth_Pr3244165891152107629_VEBTi @ ( enumerate_VEBT_VEBTi @ N2 @ Xs2 ) @ M )
        = ( produc2649746096677893406_VEBTi @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBTi @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7952_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_P6730324909620535719T_VEBT,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_s4764337671732037139T_VEBT @ Xs2 ) )
     => ( ( nth_Pr5469784954002723455T_VEBT @ ( enumer7493320076530018694T_VEBT @ N2 @ Xs2 ) @ M )
        = ( produc1750349459881913976T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_Pr2419613052044807976T_VEBT @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7953_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_num,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_num @ Xs2 ) )
     => ( ( nth_Pr8326237132889035090at_num @ ( enumerate_num @ N2 @ Xs2 ) @ M )
        = ( product_Pair_nat_num @ ( plus_plus_nat @ N2 @ M ) @ ( nth_num @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7954_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_real,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_real @ Xs2 ) )
     => ( ( nth_Pr7767817059697521252t_real @ ( enumerate_real @ N2 @ Xs2 ) @ M )
        = ( produc7837566107596912789t_real @ ( plus_plus_nat @ N2 @ M ) @ ( nth_real @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7955_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_o,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N2 @ Xs2 ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N2 @ M ) @ ( nth_o @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7956_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs2 ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7957_nth__enumerate__eq,axiom,
    ! [M: nat,Xs2: list_int,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs2 ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs2 @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_7958_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [X2: nat,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( ( vEBT_V739175172307565963ildupi @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_V254170901696579886pi_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
             => ~ ( accp_nat @ vEBT_V254170901696579886pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
               => ~ ( accp_nat @ vEBT_V254170901696579886pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                            @ ^ [TreeList: list_VEBT_VEBTi] :
                                ( heap_T7982501707604696571_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ ^ [Uu: product_unit] :
                                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                    @ ^ [Trees: array_VEBT_VEBTi] :
                                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                        @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V739175172307565963ildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                            @ ^ [TreeList: list_VEBT_VEBTi] :
                                ( heap_T7982501707604696571_VEBTi
                                @ ( refine_Imp_assert
                                  @ ( ( size_s7982070591426661849_VEBTi @ TreeList )
                                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                                @ ^ [Uu: product_unit] :
                                    ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                    @ ^ [Trees: array_VEBT_VEBTi] :
                                        ( heap_T1006145433769338483_VEBTi @ ( vEBT_V739175172307565963ildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V254170901696579886pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
thf(fact_7959_VEBTi_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( size_size_VEBT_VEBTi @ ( vEBT_Nodei @ X11 @ X122 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ size_size_VEBT_VEBTi @ X13 ) @ ( size_size_VEBT_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size(3)
thf(fact_7960_vebt__buildupi_Opelims,axiom,
    ! [X2: nat,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( ( vEBT_vebt_buildupi @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_v1230518104690509829pi_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
             => ~ ( accp_nat @ vEBT_v1230518104690509829pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( heap_T3630416162098727440_VEBTi @ ( vEBT_Leafi @ $false @ $false ) ) )
               => ~ ( accp_nat @ vEBT_v1230518104690509829pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                            @ ^ [TreeList: list_VEBT_VEBTi] :
                                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                @ ^ [Trees: array_VEBT_VEBTi] :
                                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                    @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( heap_T5877712393672139267_VEBTi @ ( vEBT_V1859673955506687831_VEBTi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildupi @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                            @ ^ [TreeList: list_VEBT_VEBTi] :
                                ( heap_T5099337393651448672_VEBTi @ ( array_615059503499738533_VEBTi @ TreeList )
                                @ ^ [Trees: array_VEBT_VEBTi] :
                                    ( heap_T1006145433769338483_VEBTi @ ( vEBT_vebt_buildupi @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                    @ ^ [Summary2: vEBT_VEBTi] : ( heap_T3630416162098727440_VEBTi @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ Trees @ Summary2 ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v1230518104690509829pi_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
thf(fact_7961_sin__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X2 )
        = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_7962_cos__tan,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X2 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_7963_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [X2: real] :
      ( ( ( sqrt @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
thf(fact_7964_real__sqrt__zero,axiom,
    ( ( sqrt @ zero_zero_real )
    = zero_zero_real ) ).

% real_sqrt_zero
thf(fact_7965_real__sqrt__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% real_sqrt_less_iff
thf(fact_7966_real__sqrt__gt__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ zero_zero_real @ Y2 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_7967_real__sqrt__lt__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_7968_real__sqrt__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_7969_real__sqrt__ge__0__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ).

% real_sqrt_ge_0_iff
thf(fact_7970_real__sqrt__gt__1__iff,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y2 ) )
      = ( ord_less_real @ one_one_real @ Y2 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_7971_real__sqrt__lt__1__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_7972_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_7973_real__sqrt__abs,axiom,
    ! [X2: real] :
      ( ( sqrt @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X2 ) ) ).

% real_sqrt_abs
thf(fact_7974_real__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 ) ) ).

% real_sqrt_pow2
thf(fact_7975_real__sqrt__pow2__iff,axiom,
    ! [X2: real] :
      ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% real_sqrt_pow2_iff
thf(fact_7976_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X2: real,Y2: real,Xa: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7977_real__sqrt__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ).

% real_sqrt_less_mono
thf(fact_7978_real__sqrt__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_7979_real__sqrt__eq__zero__cancel,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ( sqrt @ X2 )
          = zero_zero_real )
       => ( X2 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_7980_real__sqrt__ge__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_7981_real__div__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
        = ( sqrt @ X2 ) ) ) ).

% real_div_sqrt
thf(fact_7982_sqrt__add__le__add__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y2 ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_7983_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_7984_real__less__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
     => ( ord_less_real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_less_rsqrt
thf(fact_7985_sqrt__le__D,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y2 )
     => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_7986_real__le__rsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 )
     => ( ord_less_eq_real @ X2 @ ( sqrt @ Y2 ) ) ) ).

% real_le_rsqrt
thf(fact_7987_real__le__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_le_lsqrt
thf(fact_7988_real__sqrt__unique,axiom,
    ! [Y2: real,X2: real] :
      ( ( ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( sqrt @ X2 )
          = Y2 ) ) ) ).

% real_sqrt_unique
thf(fact_7989_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_7990_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X2 )
     => ( Y2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_7991_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y2 )
     => ( X2 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_7992_real__sqrt__sum__squares__ge1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_7993_real__sqrt__sum__squares__ge2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq_real @ Y2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_7994_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A3: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A3 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B3 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( 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_7995_sqrt__ge__absD,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y2 ) )
     => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y2 ) ) ).

% sqrt_ge_absD
thf(fact_7996_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_7997_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_7998_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_7999_real__less__lsqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ X2 @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X2 ) @ Y2 ) ) ) ) ).

% real_less_lsqrt
thf(fact_8000_sqrt__sum__squares__le__sum,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y2 ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_8001_sqrt__even__pow2,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_8002_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y2 ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_8003_real__sqrt__ge__abs2,axiom,
    ! [Y2: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_8004_real__sqrt__ge__abs1,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_8005_ln__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( sqrt @ X2 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8006_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_8007_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_8008_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_8009_real__sqrt__power__even,axiom,
    ! [N2: nat,X2: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( sqrt @ X2 ) @ N2 )
          = ( power_power_real @ X2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_8010_arsinh__real__aux,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8011_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X2: real,Y2: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8012_arith__geo__mean__sqrt,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y2 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8013_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_8014_cos__x__y__le__one,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_8015_real__sqrt__sum__squares__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8016_arcosh__real__def,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( arcosh_real @ X2 )
        = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_8017_cos__arctan,axiom,
    ! [X2: real] :
      ( ( cos_real @ ( arctan @ X2 ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_8018_sin__arctan,axiom,
    ! [X2: real] :
      ( ( sin_real @ ( arctan @ X2 ) )
      = ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_8019_sqrt__sum__squares__half__less,axiom,
    ! [X2: real,U: real,Y2: real] :
      ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8020_sin__cos__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
     => ( ( sin_real @ X2 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_8021_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_8022_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_V9176841429113362141ildupi @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_V3352910403632780892pi_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2 = one_one_int )
             => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2 = one_one_int )
               => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) @ ( plus_plus_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( vEBT_V9176841429113362141ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V3352910403632780892pi_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
thf(fact_8023_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8646137997579335489_i_l_d @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_V5144397997797733112_d_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
             => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_V8646137997579335489_i_l_d @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V5144397997797733112_d_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
thf(fact_8024_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V8346862874174094_d_u_p @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_V1247956027447740395_p_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus_nat @ one_one_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( vEBT_V8346862874174094_d_u_p @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( vEBT_V8346862874174094_d_u_p @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V1247956027447740395_p_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
thf(fact_8025_VEBT__internal_OTb_Opelims,axiom,
    ! [X2: nat,Y2: int] :
      ( ( ( vEBT_VEBT_Tb @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel2 @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( numeral_numeral_int @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( numeral_numeral_int @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_int @ ( vEBT_VEBT_Tb @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel2 @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
thf(fact_8026_VEBT__internal_OTb_H_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_VEBT_Tb2 @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_VEBT_Tb_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
             => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_Tb2 @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( times_times_nat @ ( vEBT_VEBT_Tb2 @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_VEBT_Tb_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
thf(fact_8027_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( vEBT_V441764108873111860ildupi @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_V2957053500504383685pi_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( suc @ zero_zero_nat ) )
             => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [N4: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ N4 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
                       => ( Y2
                          = ( suc @ ( suc @ ( suc @ ( plus_plus_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( vEBT_V441764108873111860ildupi @ ( suc @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_V2957053500504383685pi_rel @ ( suc @ ( suc @ N4 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
thf(fact_8028_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: array_VEBT_VEBTi,X14: vEBT_VEBTi] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Nodei @ X11 @ X122 @ X13 @ X14 ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( size_a6397454172108246045_VEBTi @ vEBT_size_VEBTi @ X13 ) @ ( vEBT_size_VEBTi @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).

% VEBTi.size_gen(1)
thf(fact_8029_obtain__set__succ,axiom,
    ! [X2: nat,Z: nat,A4: set_nat,B6: set_nat] :
      ( ( ord_less_nat @ X2 @ Z )
     => ( ( vEBT_VEBT_max_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ B6 )
         => ( ( A4 = B6 )
           => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A4 @ X2 @ X_1 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_8030_obtain__set__pred,axiom,
    ! [Z: nat,X2: nat,A4: set_nat] :
      ( ( ord_less_nat @ Z @ X2 )
     => ( ( vEBT_VEBT_min_in_set @ A4 @ Z )
       => ( ( finite_finite_nat @ A4 )
         => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A4 @ X2 @ X_1 ) ) ) ) ).

% obtain_set_pred
thf(fact_8031_bezw_Opelims,axiom,
    ! [X2: nat,Xa: nat,Y2: product_prod_int_int] :
      ( ( ( bezw @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y2
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y2
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_8032_set__vebt__finite,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).

% set_vebt_finite
thf(fact_8033_pred__none__empty,axiom,
    ! [Xs2: set_nat,A3: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A3 @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs2 )
              & ( ord_less_nat @ X4 @ A3 ) ) ) ) ).

% pred_none_empty
thf(fact_8034_succ__none__empty,axiom,
    ! [Xs2: set_nat,A3: nat] :
      ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A3 @ X_1 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X4: nat] :
              ( ( member_nat @ X4 @ Xs2 )
              & ( ord_less_nat @ A3 @ X4 ) ) ) ) ).

% succ_none_empty
thf(fact_8035_infinite__Icc__iff,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) ) )
      = ( ord_less_rat @ A3 @ B3 ) ) ).

% infinite_Icc_iff
thf(fact_8036_infinite__Icc__iff,axiom,
    ! [A3: real,B3: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) )
      = ( ord_less_real @ A3 @ B3 ) ) ).

% infinite_Icc_iff
thf(fact_8037_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( summable_complex
        @ ^ [R6: nat] : ( if_complex @ ( member_nat @ R6 @ A4 ) @ ( F @ R6 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_8038_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( summable_real
        @ ^ [R6: nat] : ( if_real @ ( member_nat @ R6 @ A4 ) @ ( F @ R6 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_8039_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( summable_nat
        @ ^ [R6: nat] : ( if_nat @ ( member_nat @ R6 @ A4 ) @ ( F @ R6 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_8040_summable__If__finite__set,axiom,
    ! [A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( summable_int
        @ ^ [R6: nat] : ( if_int @ ( member_nat @ R6 @ A4 ) @ ( F @ R6 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_8041_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R6: nat] : ( if_complex @ ( P @ R6 ) @ ( F @ R6 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_8042_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R6: nat] : ( if_real @ ( P @ R6 ) @ ( F @ R6 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_8043_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R6: nat] : ( if_nat @ ( P @ R6 ) @ ( F @ R6 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_8044_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R6: nat] : ( if_int @ ( P @ R6 ) @ ( F @ R6 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_8045_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N11: set_nat] :
        ? [M3: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N11 )
         => ( ord_less_nat @ X @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_8046_bounded__nat__set__is__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N3 )
         => ( ord_less_nat @ X3 @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% bounded_nat_set_is_finite
thf(fact_8047_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N11: set_nat] :
        ? [M3: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N11 )
         => ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_8048_finite__if__eq__beyond__finite,axiom,
    ! [S4: set_int,S5: set_int] :
      ( ( finite_finite_int @ S4 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [S2: set_int] :
              ( ( minus_minus_set_int @ S2 @ S4 )
              = ( minus_minus_set_int @ S5 @ S4 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8049_finite__if__eq__beyond__finite,axiom,
    ! [S4: set_complex,S5: set_complex] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [S2: set_complex] :
              ( ( minus_811609699411566653omplex @ S2 @ S4 )
              = ( minus_811609699411566653omplex @ S5 @ S4 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8050_finite__if__eq__beyond__finite,axiom,
    ! [S4: set_Code_integer,S5: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [S2: set_Code_integer] :
              ( ( minus_2355218937544613996nteger @ S2 @ S4 )
              = ( minus_2355218937544613996nteger @ S5 @ S4 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8051_finite__if__eq__beyond__finite,axiom,
    ! [S4: set_nat,S5: set_nat] :
      ( ( finite_finite_nat @ S4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [S2: set_nat] :
              ( ( minus_minus_set_nat @ S2 @ S4 )
              = ( minus_minus_set_nat @ S5 @ S4 ) ) ) ) ) ).

% finite_if_eq_beyond_finite
thf(fact_8052_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_8053_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_8054_finite__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8055_finite__lists__length__eq,axiom,
    ! [A4: set_Pr5949110396991348497n_assn,N2: nat] :
      ( ( finite5137929494490007386n_assn @ A4 )
     => ( finite1351478129840809056n_assn
        @ ( collec8177951099088521122n_assn
          @ ^ [Xs: list_P8527749157015355191n_assn] :
              ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ A4 )
              & ( ( size_s6829681357464350627n_assn @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8056_finite__lists__length__eq,axiom,
    ! [A4: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ( size_s3451745648224563538omplex @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8057_finite__lists__length__eq,axiom,
    ! [A4: set_Code_integer,N2: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A4 )
              & ( ( size_s3445333598471063425nteger @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8058_finite__lists__length__eq,axiom,
    ! [A4: set_real,N2: nat] :
      ( ( finite_finite_real @ A4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
              & ( ( size_size_list_real @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8059_finite__lists__length__eq,axiom,
    ! [A4: set_o,N2: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ( size_size_list_o @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8060_finite__lists__length__eq,axiom,
    ! [A4: set_int,N2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ( size_size_list_int @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8061_finite__lists__length__eq,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ( size_size_list_nat @ Xs )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_8062_infinite__Icc,axiom,
    ! [A3: rat,B3: rat] :
      ( ( ord_less_rat @ A3 @ B3 )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A3 @ B3 ) ) ) ).

% infinite_Icc
thf(fact_8063_infinite__Icc,axiom,
    ! [A3: real,B3: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A3 @ B3 ) ) ) ).

% infinite_Icc
thf(fact_8064_finite__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8065_finite__lists__length__le,axiom,
    ! [A4: set_Pr5949110396991348497n_assn,N2: nat] :
      ( ( finite5137929494490007386n_assn @ A4 )
     => ( finite1351478129840809056n_assn
        @ ( collec8177951099088521122n_assn
          @ ^ [Xs: list_P8527749157015355191n_assn] :
              ( ( ord_le171416862856029873n_assn @ ( set_Pr1139785259514867910n_assn @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s6829681357464350627n_assn @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8066_finite__lists__length__le,axiom,
    ! [A4: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8067_finite__lists__length__le,axiom,
    ! [A4: set_Code_integer,N2: nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite1283093830868386564nteger
        @ ( collec3483841146883906114nteger
          @ ^ [Xs: list_Code_integer] :
              ( ( ord_le7084787975880047091nteger @ ( set_Code_integer2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s3445333598471063425nteger @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8068_finite__lists__length__le,axiom,
    ! [A4: set_real,N2: nat] :
      ( ( finite_finite_real @ A4 )
     => ( finite306553202115118035t_real
        @ ( collect_list_real
          @ ^ [Xs: list_real] :
              ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_real @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8069_finite__lists__length__le,axiom,
    ! [A4: set_o,N2: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8070_finite__lists__length__le,axiom,
    ! [A4: set_int,N2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8071_finite__lists__length__le,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_8072_summable__finite,axiom,
    ! [N3: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_complex ) )
       => ( summable_complex @ F ) ) ) ).

% summable_finite
thf(fact_8073_summable__finite,axiom,
    ! [N3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_8074_summable__finite,axiom,
    ! [N3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_8075_summable__finite,axiom,
    ! [N3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_8076_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_8077_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_8078_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z3: real] :
              ( ( power_power_real @ Z3 @ N2 )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_8079_finite__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_8080_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBTi @ ( vEBT_Leafi @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBTi.size_gen(2)
thf(fact_8081_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_8082_finite__Collect__less__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).

% finite_Collect_less_nat
thf(fact_8083_finite__Collect__subsets,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( finite6197958912794628473et_int
        @ ( collect_set_int
          @ ^ [B7: set_int] : ( ord_less_eq_set_int @ B7 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_8084_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B7: set_complex] : ( ord_le211207098394363844omplex @ B7 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_8085_finite__Collect__subsets,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( finite6931041176100689706nteger
        @ ( collec574505750873337192nteger
          @ ^ [B7: set_Code_integer] : ( ord_le7084787975880047091nteger @ B7 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_8086_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B7: set_nat] : ( ord_less_eq_set_nat @ B7 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_8087_finite__induct__select,axiom,
    ! [S4: set_VEBT_VEBT,P: set_VEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [T5: set_VEBT_VEBT] :
              ( ( ord_le3480810397992357184T_VEBT @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( minus_5127226145743854075T_VEBT @ S4 @ T5 ) )
                    & ( P @ ( insert_VEBT_VEBT @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8088_finite__induct__select,axiom,
    ! [S4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T5: set_complex] :
              ( ( ord_less_set_complex @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: complex] :
                    ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ S4 @ T5 ) )
                    & ( P @ ( insert_complex @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8089_finite__induct__select,axiom,
    ! [S4: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [T5: set_Code_integer] :
              ( ( ord_le1307284697595431911nteger @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ ( minus_2355218937544613996nteger @ S4 @ T5 ) )
                    & ( P @ ( insert_Code_integer @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8090_finite__induct__select,axiom,
    ! [S4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T5: set_real] :
              ( ( ord_less_set_real @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: real] :
                    ( ( member_real @ X4 @ ( minus_minus_set_real @ S4 @ T5 ) )
                    & ( P @ ( insert_real @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8091_finite__induct__select,axiom,
    ! [S4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T5: set_int] :
              ( ( ord_less_set_int @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: int] :
                    ( ( member_int @ X4 @ ( minus_minus_set_int @ S4 @ T5 ) )
                    & ( P @ ( insert_int @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8092_finite__induct__select,axiom,
    ! [S4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T5: set_nat] :
              ( ( ord_less_set_nat @ T5 @ S4 )
             => ( ( P @ T5 )
               => ? [X4: nat] :
                    ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S4 @ T5 ) )
                    & ( P @ ( insert_nat @ X4 @ T5 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_induct_select
thf(fact_8093_set__encode__insert,axiom,
    ! [A4: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ N2 @ A4 )
       => ( ( nat_set_encode @ ( insert_nat @ N2 @ A4 ) )
          = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).

% set_encode_insert
thf(fact_8094_finite__Collect__disjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_8095_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_8096_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_8097_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_8098_finite__Collect__disjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        & ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_8099_finite__Collect__conjI,axiom,
    ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
      ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
        | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8100_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8101_finite__Collect__conjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ( finite_finite_int @ ( collect_int @ P ) )
        | ( finite_finite_int @ ( collect_int @ Q ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8102_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8103_finite__Collect__conjI,axiom,
    ! [P: code_integer > $o,Q: code_integer > $o] :
      ( ( ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
        | ( finite6017078050557962740nteger @ ( collect_Code_integer @ Q ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8104_finite__interval__int1,axiom,
    ! [A3: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A3 @ I4 )
            & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int1
thf(fact_8105_finite__interval__int4,axiom,
    ! [A3: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A3 @ I4 )
            & ( ord_less_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int4
thf(fact_8106_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_8107_finite__interval__int3,axiom,
    ! [A3: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_int @ A3 @ I4 )
            & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int3
thf(fact_8108_finite__interval__int2,axiom,
    ! [A3: int,B3: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I4: int] :
            ( ( ord_less_eq_int @ A3 @ I4 )
            & ( ord_less_int @ I4 @ B3 ) ) ) ) ).

% finite_interval_int2
thf(fact_8109_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_8110_finite__maxlen,axiom,
    ! [M8: set_list_real] :
      ( ( finite306553202115118035t_real @ M8 )
     => ? [N4: nat] :
        ! [X4: list_real] :
          ( ( member_list_real @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_real @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_8111_finite__maxlen,axiom,
    ! [M8: set_list_o] :
      ( ( finite_finite_list_o @ M8 )
     => ? [N4: nat] :
        ! [X4: list_o] :
          ( ( member_list_o @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_8112_finite__maxlen,axiom,
    ! [M8: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M8 )
     => ? [N4: nat] :
        ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_8113_finite__maxlen,axiom,
    ! [M8: set_list_int] :
      ( ( finite3922522038869484883st_int @ M8 )
     => ? [N4: nat] :
        ! [X4: list_int] :
          ( ( member_list_int @ X4 @ M8 )
         => ( ord_less_nat @ ( size_size_list_int @ X4 ) @ N4 ) ) ) ).

% finite_maxlen
thf(fact_8114_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_8115_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( nat_set_encode @ A4 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_8116_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_nat,R4: vEBT_VEBT > nat > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B6 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8117_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B6: set_nat,R4: real > nat > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B6 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8118_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_int,R4: vEBT_VEBT > int > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_int @ B6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B6 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8119_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B6: set_int,R4: real > int > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite_finite_int @ B6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B6 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8120_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_complex,R4: vEBT_VEBT > complex > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa3: complex] :
                  ( ( member_complex @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B6 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8121_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B6: set_complex,R4: real > complex > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa3: complex] :
                  ( ( member_complex @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: complex] :
              ( ( member_complex @ X3 @ B6 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8122_pigeonhole__infinite__rel,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_Code_integer,R4: vEBT_VEBT > code_integer > $o] :
      ( ~ ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite6017078050557962740nteger @ B6 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B6 )
              & ~ ( finite5795047828879050333T_VEBT
                  @ ( collect_VEBT_VEBT
                    @ ^ [A2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8123_pigeonhole__infinite__rel,axiom,
    ! [A4: set_real,B6: set_Code_integer,R4: real > code_integer > $o] :
      ( ~ ( finite_finite_real @ A4 )
     => ( ( finite6017078050557962740nteger @ B6 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ B6 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A2: real] :
                        ( ( member_real @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8124_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B6: set_nat,R4: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: nat] :
              ( ( member_nat @ X3 @ B6 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8125_pigeonhole__infinite__rel,axiom,
    ! [A4: set_nat,B6: set_int,R4: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_finite_int @ B6 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ B6 )
                  & ( R4 @ X3 @ Xa3 ) ) )
         => ? [X3: int] :
              ( ( member_int @ X3 @ B6 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A2: nat] :
                        ( ( member_nat @ A2 @ A4 )
                        & ( R4 @ A2 @ X3 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_8126_not__finite__existsD,axiom,
    ! [P: product_prod_int_int > $o] :
      ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
     => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_8127_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_8128_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_8129_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_1: complex] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_8130_not__finite__existsD,axiom,
    ! [P: code_integer > $o] :
      ( ~ ( finite6017078050557962740nteger @ ( collect_Code_integer @ P ) )
     => ? [X_1: code_integer] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_8131_finite__has__maximal2,axiom,
    ! [A4: set_real,A3: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A3 @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ A3 @ X3 )
            & ! [Xa3: real] :
                ( ( member_real @ Xa3 @ A4 )
               => ( ( ord_less_eq_real @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8132_finite__has__maximal2,axiom,
    ! [A4: set_Code_integer,A3: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A3 @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ A3 @ X3 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8133_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A3 @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ A3 @ X3 )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8134_finite__has__maximal2,axiom,
    ! [A4: set_rat,A3: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A3 @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ A3 @ X3 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8135_finite__has__maximal2,axiom,
    ! [A4: set_num,A3: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A3 @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ A3 @ X3 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8136_finite__has__maximal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ A3 @ X3 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8137_finite__has__maximal2,axiom,
    ! [A4: set_int,A3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A3 @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ A3 @ X3 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_8138_finite__has__minimal2,axiom,
    ! [A4: set_real,A3: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A3 @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ X3 @ A3 )
            & ! [Xa3: real] :
                ( ( member_real @ Xa3 @ A4 )
               => ( ( ord_less_eq_real @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8139_finite__has__minimal2,axiom,
    ! [A4: set_Code_integer,A3: code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( member_Code_integer @ A3 @ A4 )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ( ord_le3102999989581377725nteger @ X3 @ A3 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8140_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A3: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A3 @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ X3 @ A3 )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8141_finite__has__minimal2,axiom,
    ! [A4: set_rat,A3: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A3 @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ X3 @ A3 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8142_finite__has__minimal2,axiom,
    ! [A4: set_num,A3: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A3 @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ X3 @ A3 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8143_finite__has__minimal2,axiom,
    ! [A4: set_nat,A3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A3 @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ X3 @ A3 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8144_finite__has__minimal2,axiom,
    ! [A4: set_int,A3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A3 @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ X3 @ A3 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_8145_finite__psubset__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [A8: set_nat] :
            ( ( finite_finite_nat @ A8 )
           => ( ! [B8: set_nat] :
                  ( ( ord_less_set_nat @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_8146_finite__psubset__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [A8: set_int] :
            ( ( finite_finite_int @ A8 )
           => ( ! [B8: set_int] :
                  ( ( ord_less_set_int @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_8147_finite__psubset__induct,axiom,
    ! [A4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [A8: set_complex] :
            ( ( finite3207457112153483333omplex @ A8 )
           => ( ! [B8: set_complex] :
                  ( ( ord_less_set_complex @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_8148_finite__psubset__induct,axiom,
    ! [A4: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [A8: set_Code_integer] :
            ( ( finite6017078050557962740nteger @ A8 )
           => ( ! [B8: set_Code_integer] :
                  ( ( ord_le1307284697595431911nteger @ B8 @ A8 )
                 => ( P @ B8 ) )
             => ( P @ A8 ) ) )
       => ( P @ A4 ) ) ) ).

% finite_psubset_induct
thf(fact_8149_even__set__encode__iff,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A4 ) )
        = ( ~ ( member_nat @ zero_zero_nat @ A4 ) ) ) ) ).

% even_set_encode_iff
thf(fact_8150_finite__has__maximal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8151_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ! [Xa3: real] :
                ( ( member_real @ Xa3 @ A4 )
               => ( ( ord_less_eq_real @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8152_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8153_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8154_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8155_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8156_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_8157_finite__has__minimal,axiom,
    ! [A4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
            & ! [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ A4 )
               => ( ( ord_le3102999989581377725nteger @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8158_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ! [Xa3: real] :
                ( ( member_real @ Xa3 @ A4 )
               => ( ( ord_less_eq_real @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8159_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ! [Xa3: set_nat] :
                ( ( member_set_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8160_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa3: rat] :
                ( ( member_rat @ Xa3 @ A4 )
               => ( ( ord_less_eq_rat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8161_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa3: num] :
                ( ( member_num @ Xa3 @ A4 )
               => ( ( ord_less_eq_num @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8162_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa3: nat] :
                ( ( member_nat @ Xa3 @ A4 )
               => ( ( ord_less_eq_nat @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8163_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa3: int] :
                ( ( member_int @ Xa3 @ A4 )
               => ( ( ord_less_eq_int @ Xa3 @ X3 )
                 => ( X3 = Xa3 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_8164_diff__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,N3: product_prod_int_int > nat] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8165_diff__preserves__multiset,axiom,
    ! [M8: nat > nat,N3: nat > nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8166_diff__preserves__multiset,axiom,
    ! [M8: int > nat,N3: int > nat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8167_diff__preserves__multiset,axiom,
    ! [M8: complex > nat,N3: complex > nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8168_diff__preserves__multiset,axiom,
    ! [M8: code_integer > nat,N3: code_integer > nat] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( M8 @ X ) @ ( N3 @ X ) ) ) ) ) ) ).

% diff_preserves_multiset
thf(fact_8169_add__mset__in__multiset,axiom,
    ! [M8: product_prod_int_int > nat,A3: product_prod_int_int] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A3 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8170_add__mset__in__multiset,axiom,
    ! [M8: nat > nat,A3: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A3 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8171_add__mset__in__multiset,axiom,
    ! [M8: int > nat,A3: int] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A3 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8172_add__mset__in__multiset,axiom,
    ! [M8: complex > nat,A3: complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A3 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8173_add__mset__in__multiset,axiom,
    ! [M8: code_integer > nat,A3: code_integer] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A3 ) @ ( suc @ ( M8 @ X ) ) @ ( M8 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_8174_finite__linorder__min__induct,axiom,
    ! [A4: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A8 )
                   => ( ord_le6747313008572928689nteger @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8175_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A8 )
                   => ( ord_less_real @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8176_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A8 )
                   => ( ord_less_rat @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8177_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A8 )
                   => ( ord_less_num @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8178_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A8 )
                   => ( ord_less_nat @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8179_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A8 )
                   => ( ord_less_int @ B @ X4 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_8180_finite__linorder__max__induct,axiom,
    ! [A4: set_Code_integer,P: set_Code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [B: code_integer,A8: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ A8 )
             => ( ! [X4: code_integer] :
                    ( ( member_Code_integer @ X4 @ A8 )
                   => ( ord_le6747313008572928689nteger @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_Code_integer @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8181_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X4: real] :
                    ( ( member_real @ X4 @ A8 )
                   => ( ord_less_real @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8182_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X4: rat] :
                    ( ( member_rat @ X4 @ A8 )
                   => ( ord_less_rat @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8183_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X4: num] :
                    ( ( member_num @ X4 @ A8 )
                   => ( ord_less_num @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8184_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ A8 )
                   => ( ord_less_nat @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8185_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X4: int] :
                    ( ( member_int @ X4 @ A8 )
                   => ( ord_less_int @ X4 @ B ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_8186_finite__ranking__induct,axiom,
    ! [S4: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8187_finite__ranking__induct,axiom,
    ! [S4: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8188_finite__ranking__induct,axiom,
    ! [S4: set_Code_integer,P: set_Code_integer > $o,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8189_finite__ranking__induct,axiom,
    ! [S4: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_real @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8190_finite__ranking__induct,axiom,
    ! [S4: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X3: nat,S6: set_nat] :
              ( ( finite_finite_nat @ S6 )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_nat @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8191_finite__ranking__induct,axiom,
    ! [S4: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X3: int,S6: set_int] :
              ( ( finite_finite_int @ S6 )
             => ( ! [Y4: int] :
                    ( ( member_int @ Y4 @ S6 )
                   => ( ord_less_eq_rat @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_int @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8192_finite__ranking__induct,axiom,
    ! [S4: set_VEBT_VEBT,P: set_VEBT_VEBT > $o,F: vEBT_VEBT > num] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( P @ bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT,S6: set_VEBT_VEBT] :
              ( ( finite5795047828879050333T_VEBT @ S6 )
             => ( ! [Y4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_VEBT_VEBT @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8193_finite__ranking__induct,axiom,
    ! [S4: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X3: complex,S6: set_complex] :
              ( ( finite3207457112153483333omplex @ S6 )
             => ( ! [Y4: complex] :
                    ( ( member_complex @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_complex @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8194_finite__ranking__induct,axiom,
    ! [S4: set_Code_integer,P: set_Code_integer > $o,F: code_integer > num] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( P @ bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer,S6: set_Code_integer] :
              ( ( finite6017078050557962740nteger @ S6 )
             => ( ! [Y4: code_integer] :
                    ( ( member_Code_integer @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_Code_integer @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8195_finite__ranking__induct,axiom,
    ! [S4: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X3: real,S6: set_real] :
              ( ( finite_finite_real @ S6 )
             => ( ! [Y4: real] :
                    ( ( member_real @ Y4 @ S6 )
                   => ( ord_less_eq_num @ ( F @ Y4 ) @ ( F @ X3 ) ) )
               => ( ( P @ S6 )
                 => ( P @ ( insert_real @ X3 @ S6 ) ) ) ) )
         => ( P @ S4 ) ) ) ) ).

% finite_ranking_induct
thf(fact_8196_ex__min__if__finite,axiom,
    ! [S4: set_Code_integer] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( S4 != bot_bo3990330152332043303nteger )
       => ? [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ S4 )
            & ~ ? [Xa3: code_integer] :
                  ( ( member_Code_integer @ Xa3 @ S4 )
                  & ( ord_le6747313008572928689nteger @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8197_ex__min__if__finite,axiom,
    ! [S4: set_real] :
      ( ( finite_finite_real @ S4 )
     => ( ( S4 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ S4 )
            & ~ ? [Xa3: real] :
                  ( ( member_real @ Xa3 @ S4 )
                  & ( ord_less_real @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8198_ex__min__if__finite,axiom,
    ! [S4: set_rat] :
      ( ( finite_finite_rat @ S4 )
     => ( ( S4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ S4 )
            & ~ ? [Xa3: rat] :
                  ( ( member_rat @ Xa3 @ S4 )
                  & ( ord_less_rat @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8199_ex__min__if__finite,axiom,
    ! [S4: set_num] :
      ( ( finite_finite_num @ S4 )
     => ( ( S4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ S4 )
            & ~ ? [Xa3: num] :
                  ( ( member_num @ Xa3 @ S4 )
                  & ( ord_less_num @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8200_ex__min__if__finite,axiom,
    ! [S4: set_nat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( S4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ S4 )
            & ~ ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ S4 )
                  & ( ord_less_nat @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8201_ex__min__if__finite,axiom,
    ! [S4: set_int] :
      ( ( finite_finite_int @ S4 )
     => ( ( S4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ S4 )
            & ~ ? [Xa3: int] :
                  ( ( member_int @ Xa3 @ S4 )
                  & ( ord_less_int @ Xa3 @ X3 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_8202_infinite__growing,axiom,
    ! [X9: set_Code_integer] :
      ( ( X9 != bot_bo3990330152332043303nteger )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ X9 )
           => ? [Xa3: code_integer] :
                ( ( member_Code_integer @ Xa3 @ X9 )
                & ( ord_le6747313008572928689nteger @ X3 @ Xa3 ) ) )
       => ~ ( finite6017078050557962740nteger @ X9 ) ) ) ).

% infinite_growing
thf(fact_8203_infinite__growing,axiom,
    ! [X9: set_real] :
      ( ( X9 != bot_bot_set_real )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ X9 )
           => ? [Xa3: real] :
                ( ( member_real @ Xa3 @ X9 )
                & ( ord_less_real @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_real @ X9 ) ) ) ).

% infinite_growing
thf(fact_8204_infinite__growing,axiom,
    ! [X9: set_rat] :
      ( ( X9 != bot_bot_set_rat )
     => ( ! [X3: rat] :
            ( ( member_rat @ X3 @ X9 )
           => ? [Xa3: rat] :
                ( ( member_rat @ Xa3 @ X9 )
                & ( ord_less_rat @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_rat @ X9 ) ) ) ).

% infinite_growing
thf(fact_8205_infinite__growing,axiom,
    ! [X9: set_num] :
      ( ( X9 != bot_bot_set_num )
     => ( ! [X3: num] :
            ( ( member_num @ X3 @ X9 )
           => ? [Xa3: num] :
                ( ( member_num @ Xa3 @ X9 )
                & ( ord_less_num @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_num @ X9 ) ) ) ).

% infinite_growing
thf(fact_8206_infinite__growing,axiom,
    ! [X9: set_nat] :
      ( ( X9 != bot_bot_set_nat )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ X9 )
           => ? [Xa3: nat] :
                ( ( member_nat @ Xa3 @ X9 )
                & ( ord_less_nat @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_nat @ X9 ) ) ) ).

% infinite_growing
thf(fact_8207_infinite__growing,axiom,
    ! [X9: set_int] :
      ( ( X9 != bot_bot_set_int )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ X9 )
           => ? [Xa3: int] :
                ( ( member_int @ Xa3 @ X9 )
                & ( ord_less_int @ X3 @ Xa3 ) ) )
       => ~ ( finite_finite_int @ X9 ) ) ) ).

% infinite_growing
thf(fact_8208_filter__preserves__multiset,axiom,
    ! [M8: product_prod_int_int > nat,P: product_prod_int_int > $o] :
      ( ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite2998713641127702882nt_int
        @ ( collec213857154873943460nt_int
          @ ^ [X: product_prod_int_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8209_filter__preserves__multiset,axiom,
    ! [M8: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8210_filter__preserves__multiset,axiom,
    ! [M8: int > nat,P: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8211_filter__preserves__multiset,axiom,
    ! [M8: complex > nat,P: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8212_filter__preserves__multiset,axiom,
    ! [M8: code_integer > nat,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( M8 @ X ) ) ) )
     => ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [X: code_integer] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M8 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_8213_infinite__int__iff__unbounded,axiom,
    ! [S4: set_int] :
      ( ( ~ ( finite_finite_int @ S4 ) )
      = ( ! [M3: int] :
          ? [N: int] :
            ( ( ord_less_int @ M3 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S4 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_8214_infinite__int__iff__unbounded__le,axiom,
    ! [S4: set_int] :
      ( ( ~ ( finite_finite_int @ S4 ) )
      = ( ! [M3: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S4 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_8215_cos__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_8216_arcsin__0,axiom,
    ( ( arcsin @ zero_zero_real )
    = zero_zero_real ) ).

% arcsin_0
thf(fact_8217_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_8218_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_8219_arcsin__less__arcsin,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_8220_arcsin__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% arcsin_less_mono
thf(fact_8221_cos__arcsin__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X2 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_8222_infinite__nat__iff__unbounded,axiom,
    ! [S4: set_nat] :
      ( ( ~ ( finite_finite_nat @ S4 ) )
      = ( ! [M3: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( member_nat @ N @ S4 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_8223_unbounded__k__infinite,axiom,
    ! [K: nat,S4: set_nat] :
      ( ! [M4: nat] :
          ( ( ord_less_nat @ K @ M4 )
         => ? [N5: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ( member_nat @ N5 @ S4 ) ) )
     => ~ ( finite_finite_nat @ S4 ) ) ).

% unbounded_k_infinite
thf(fact_8224_infinite__nat__iff__unbounded__le,axiom,
    ! [S4: set_nat] :
      ( ( ~ ( finite_finite_nat @ S4 ) )
      = ( ! [M3: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( member_nat @ N @ S4 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_8225_arcsin__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_8226_arcsin__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).

% arcsin_lbound
thf(fact_8227_arcsin__ubound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_8228_arcsin__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_8229_arcsin__sin,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X2 ) )
          = X2 ) ) ) ).

% arcsin_sin
thf(fact_8230_le__arcsin__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y2 @ ( arcsin @ X2 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y2 ) @ X2 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_8231_arcsin__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y2 )
         => ( ( ord_less_eq_real @ Y2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y2 )
              = ( ord_less_eq_real @ X2 @ ( sin_real @ Y2 ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_8232_arcsin__pi,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin_pi
thf(fact_8233_arcsin,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y2 ) )
          & ( ord_less_eq_real @ ( arcsin @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y2 ) )
            = Y2 ) ) ) ) ).

% arcsin
thf(fact_8234_vebt__buildup_Oelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( ( X2 = zero_zero_nat )
         => ( Y2
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X2
              = ( suc @ zero_zero_nat ) )
           => ( Y2
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va3: nat] :
                ( ( X2
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y2
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_8235_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > uint32,Y2: vEBT_VEBT > uint32] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8236_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > uint32,Y2: real > uint32] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8237_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > uint32,Y2: nat > uint32] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8238_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > uint32,Y2: int > uint32] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8239_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > uint32,Y2: complex > uint32] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8240_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X2: code_integer > uint32,Y2: code_integer > uint32] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_uint32 ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_uint32 ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( times_times_uint32 @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_uint32 ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8241_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > real,Y2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8242_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8243_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8244_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_8245_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > real,Y2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8246_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > real,Y2: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8247_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > real,Y2: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8248_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > real,Y2: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8249_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X2: complex > real,Y2: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8250_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_Code_integer,X2: code_integer > real,Y2: code_integer > real] :
      ( ( finite6017078050557962740nteger
        @ ( collect_Code_integer
          @ ^ [I4: code_integer] :
              ( ( member_Code_integer @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite6017078050557962740nteger
          @ ( collect_Code_integer
            @ ^ [I4: code_integer] :
                ( ( member_Code_integer @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8251_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > rat,Y2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT
        @ ( collect_VEBT_VEBT
          @ ^ [I4: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite5795047828879050333T_VEBT
          @ ( collect_VEBT_VEBT
            @ ^ [I4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8252_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X2: real > rat,Y2: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I4: real] :
              ( ( member_real @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8253_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X2: nat > rat,Y2: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8254_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X2: int > rat,Y2: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X2 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y2 @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y2 @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_8255_ceiling__log__eq__powr__iff,axiom,
    ! [X2: real,B3: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B3 @ X2 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
            & ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8256_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X2: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_8257_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBTi > $o,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_8258_intind,axiom,
    ! [I: nat,N2: nat,P: int > $o,X2: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_8259_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X2 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) @ I ) ) ) ) ).

% intind
thf(fact_8260_powr__0,axiom,
    ! [Z: real] :
      ( ( powr_real @ zero_zero_real @ Z )
      = zero_zero_real ) ).

% powr_0
thf(fact_8261_powr__eq__0__iff,axiom,
    ! [W: real,Z: real] :
      ( ( ( powr_real @ W @ Z )
        = zero_zero_real )
      = ( W = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_8262_powr__one__eq__one,axiom,
    ! [A3: real] :
      ( ( powr_real @ one_one_real @ A3 )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_8263_replicate__eq__replicate,axiom,
    ! [M: nat,X2: vEBT_VEBT,N2: nat,Y2: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X2 )
        = ( replicate_VEBT_VEBT @ N2 @ Y2 ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X2 = Y2 ) ) ) ) ).

% replicate_eq_replicate
thf(fact_8264_length__replicate,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_8265_length__replicate,axiom,
    ! [N2: nat,X2: real] :
      ( ( size_size_list_real @ ( replicate_real @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_8266_length__replicate,axiom,
    ! [N2: nat,X2: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_8267_length__replicate,axiom,
    ! [N2: nat,X2: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_8268_length__replicate,axiom,
    ! [N2: nat,X2: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X2 ) )
      = N2 ) ).

% length_replicate
thf(fact_8269_powr__zero__eq__one,axiom,
    ! [X2: real] :
      ( ( ( X2 = zero_zero_real )
       => ( ( powr_real @ X2 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X2 != zero_zero_real )
       => ( ( powr_real @ X2 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_8270_in__set__replicate,axiom,
    ! [X2: nat,N2: nat,Y2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8271_in__set__replicate,axiom,
    ! [X2: real,N2: nat,Y2: real] :
      ( ( member_real @ X2 @ ( set_real2 @ ( replicate_real @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8272_in__set__replicate,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8273_in__set__replicate,axiom,
    ! [X2: produc6575502325842934193n_assn,N2: nat,Y2: produc6575502325842934193n_assn] :
      ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8274_in__set__replicate,axiom,
    ! [X2: vEBT_VEBT,N2: nat,Y2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y2 ) ) )
      = ( ( X2 = Y2 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8275_Bex__set__replicate,axiom,
    ! [N2: nat,A3: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A3 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_8276_Bex__set__replicate,axiom,
    ! [N2: nat,A3: real,P: real > $o] :
      ( ( ? [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ A3 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_8277_Bex__set__replicate,axiom,
    ! [N2: nat,A3: int,P: int > $o] :
      ( ( ? [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A3 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_8278_Bex__set__replicate,axiom,
    ! [N2: nat,A3: produc6575502325842934193n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ? [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ A3 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_8279_Bex__set__replicate,axiom,
    ! [N2: nat,A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A3 ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_8280_Ball__set__replicate,axiom,
    ! [N2: nat,A3: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A3 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_8281_Ball__set__replicate,axiom,
    ! [N2: nat,A3: real,P: real > $o] :
      ( ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ A3 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_8282_Ball__set__replicate,axiom,
    ! [N2: nat,A3: int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A3 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_8283_Ball__set__replicate,axiom,
    ! [N2: nat,A3: produc6575502325842934193n_assn,P: produc6575502325842934193n_assn > $o] :
      ( ( ! [X: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ A3 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_8284_Ball__set__replicate,axiom,
    ! [N2: nat,A3: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A3 ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_8285_powr__gt__zero,axiom,
    ! [X2: real,A3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A3 ) )
      = ( X2 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8286_powr__nonneg__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A3 @ X2 ) @ zero_zero_real )
      = ( A3 = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_8287_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_8288_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBTi @ ( replicate_VEBT_VEBTi @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_8289_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_8290_nth__replicate,axiom,
    ! [I: nat,N2: nat,X2: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X2 ) @ I )
        = X2 ) ) ).

% nth_replicate
thf(fact_8291_powr__less__cancel__iff,axiom,
    ! [X2: real,A3: real,B3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ X2 @ B3 ) )
        = ( ord_less_real @ A3 @ B3 ) ) ) ).

% powr_less_cancel_iff
thf(fact_8292_powr__eq__one__iff,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ one_one_real @ A3 )
     => ( ( ( powr_real @ A3 @ X2 )
          = one_one_real )
        = ( X2 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_8293_powr__one__gt__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( powr_real @ X2 @ one_one_real )
        = X2 )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% powr_one_gt_zero_iff
thf(fact_8294_powr__one,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ one_one_real )
        = X2 ) ) ).

% powr_one
thf(fact_8295_powr__le__cancel__iff,axiom,
    ! [X2: real,A3: real,B3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ X2 @ B3 ) )
        = ( ord_less_eq_real @ A3 @ B3 ) ) ) ).

% powr_le_cancel_iff
thf(fact_8296_numeral__powr__numeral__real,axiom,
    ! [M: num,N2: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8297_set__replicate,axiom,
    ! [N2: nat,X2: produc6575502325842934193n_assn] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ X2 ) )
        = ( insert5290817439147925377n_assn @ X2 @ bot_bo1176836662018730877n_assn ) ) ) ).

% set_replicate
thf(fact_8298_set__replicate,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
        = ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_8299_set__replicate,axiom,
    ! [N2: nat,X2: real] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N2 @ X2 ) )
        = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_8300_set__replicate,axiom,
    ! [N2: nat,X2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N2 @ X2 ) )
        = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_8301_set__replicate,axiom,
    ! [N2: nat,X2: int] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N2 @ X2 ) )
        = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_8302_powr__log__cancel,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( powr_real @ A3 @ ( log @ A3 @ X2 ) )
            = X2 ) ) ) ) ).

% powr_log_cancel
thf(fact_8303_log__powr__cancel,axiom,
    ! [A3: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( log @ A3 @ ( powr_real @ A3 @ Y2 ) )
          = Y2 ) ) ) ).

% log_powr_cancel
thf(fact_8304_powr__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N2 ) )
        = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_8305_map__fst__mk__fst,axiom,
    ! [K: num,L2: list_num] :
      ( ( map_Pr454908937103039467um_num @ product_fst_num_num @ ( map_nu2851882102140640437um_num @ ( product_Pair_num_num @ K ) @ L2 ) )
      = ( replicate_num @ ( size_size_list_num @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8306_map__fst__mk__fst,axiom,
    ! [K: nat,L2: list_P6730324909620535719T_VEBT] :
      ( ( map_Pr3018521781701129308BT_nat @ produc758997459209783180T_VEBT @ ( map_Pr7909041933469137955T_VEBT @ ( produc1750349459881913976T_VEBT @ K ) @ L2 ) )
      = ( replicate_nat @ ( size_s4764337671732037139T_VEBT @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8307_map__fst__mk__fst,axiom,
    ! [K: nat,L2: list_num] :
      ( ( map_Pr5956769322976601943um_nat @ product_fst_nat_num @ ( map_nu4721551698833171051at_num @ ( product_Pair_nat_num @ K ) @ L2 ) )
      = ( replicate_nat @ ( size_size_list_num @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8308_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L2: list_real] :
      ( ( map_Pr6195879527588727455T_VEBT @ produc8110914911036349469T_real @ ( map_re8618229306769252225T_real @ ( produc8117437818029410057T_real @ K ) @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8309_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L2: list_o] :
      ( ( map_Pr4868735216952053677T_VEBT @ produc4993121158135996263VEBT_o @ ( map_o_6754667662019005495VEBT_o @ ( produc8721562602347293563VEBT_o @ K ) @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8310_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L2: list_nat] :
      ( ( map_Pr1380729192516676091T_VEBT @ produc8713918199166443969BT_nat @ ( map_na4631810538828370761BT_nat @ ( produc738532404422230701BT_nat @ K ) @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8311_map__fst__mk__fst,axiom,
    ! [K: vEBT_VEBT,L2: list_int] :
      ( ( map_Pr3257657825534036127T_VEBT @ produc8711427728657393693BT_int @ ( map_in8151279748432256513BT_int @ ( produc736041933913180425BT_int @ K ) @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8312_map__fst__mk__fst,axiom,
    ! [K: nat,L2: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ K ) @ L2 ) )
      = ( replicate_nat @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8313_map__fst__mk__fst,axiom,
    ! [K: int,L2: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ ( map_in7157766398909135175nt_int @ ( product_Pair_int_int @ K ) @ L2 ) )
      = ( replicate_int @ ( size_size_list_int @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8314_map__fst__mk__fst,axiom,
    ! [K: assn,L2: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ ( map_as2373307505041272643n_assn @ ( produc118845697133431529n_assn @ K ) @ L2 ) )
      = ( replicate_assn @ ( size_size_list_assn @ L2 ) @ K ) ) ).

% map_fst_mk_fst
thf(fact_8315_map__snd__mk__snd,axiom,
    ! [K: num,L2: list_num] :
      ( ( map_Pr454908937103039467um_num @ product_snd_num_num
        @ ( map_nu2851882102140640437um_num
          @ ^ [X: num] : ( product_Pair_num_num @ X @ K )
          @ L2 ) )
      = ( replicate_num @ ( size_size_list_num @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8316_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L2: list_real] :
      ( ( map_Pr6147841162850987569T_VEBT @ produc5083336317046741121T_VEBT
        @ ( map_re7205069664741861231T_VEBT
          @ ^ [X: real] : ( produc6931449550656315951T_VEBT @ X @ K )
          @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8317_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L2: list_o] :
      ( ( map_Pr7652832201708611317T_VEBT @ produc7938581201502569057T_VEBT
        @ ( map_o_8925299737569714927T_VEBT
          @ ^ [X: $o] : ( produc2982872950893828659T_VEBT @ X @ K )
          @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8318_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L2: list_nat] :
      ( ( map_Pr8570210702748812117T_VEBT @ produc8172668247895388509T_VEBT
        @ ( map_na3584885621601055599T_VEBT
          @ ^ [X: nat] : ( produc599794634098209291T_VEBT @ X @ K )
          @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8319_map__snd__mk__snd,axiom,
    ! [K: produc4813437837504472865T_VEBT,L2: list_nat] :
      ( ( map_Pr3651661567648760725T_VEBT @ produc2084898568784432842T_VEBT
        @ ( map_na3322839687800594908T_VEBT
          @ ^ [X: nat] : ( produc1750349459881913976T_VEBT @ X @ K )
          @ L2 ) )
      = ( replic862811827245231841T_VEBT @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8320_map__snd__mk__snd,axiom,
    ! [K: num,L2: list_nat] :
      ( ( map_Pr2514101109132380577um_num @ product_snd_nat_num
        @ ( map_na8006665559001981237at_num
          @ ^ [X: nat] : ( product_Pair_nat_num @ X @ K )
          @ L2 ) )
      = ( replicate_num @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8321_map__snd__mk__snd,axiom,
    ! [K: vEBT_VEBT,L2: list_int] :
      ( ( map_Pr1314269154781486001T_VEBT @ produc1678900780639429121T_VEBT
        @ ( map_in4788438383458178671T_VEBT
          @ ^ [X: int] : ( produc3329399203697025711T_VEBT @ X @ K )
          @ L2 ) )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8322_map__snd__mk__snd,axiom,
    ! [K: nat,L2: list_nat] :
      ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ K )
          @ L2 ) )
      = ( replicate_nat @ ( size_size_list_nat @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8323_map__snd__mk__snd,axiom,
    ! [K: int,L2: list_int] :
      ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int
        @ ( map_in7157766398909135175nt_int
          @ ^ [X: int] : ( product_Pair_int_int @ X @ K )
          @ L2 ) )
      = ( replicate_int @ ( size_size_list_int @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8324_map__snd__mk__snd,axiom,
    ! [K: assn,L2: list_assn] :
      ( ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn
        @ ( map_as2373307505041272643n_assn
          @ ^ [X: assn] : ( produc118845697133431529n_assn @ X @ K )
          @ L2 ) )
      = ( replicate_assn @ ( size_size_list_assn @ L2 ) @ K ) ) ).

% map_snd_mk_snd
thf(fact_8325_square__powr__half,axiom,
    ! [X2: real] :
      ( ( powr_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X2 ) ) ).

% square_powr_half
thf(fact_8326_powr__less__mono2__neg,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y2 )
         => ( ord_less_real @ ( powr_real @ Y2 @ A3 ) @ ( powr_real @ X2 @ A3 ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_8327_powr__non__neg,axiom,
    ! [A3: real,X2: real] :
      ~ ( ord_less_real @ ( powr_real @ A3 @ X2 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8328_powr__mono2,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).

% powr_mono2
thf(fact_8329_powr__ge__pzero,axiom,
    ! [X2: real,Y2: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y2 ) ) ).

% powr_ge_pzero
thf(fact_8330_powr__less__cancel,axiom,
    ! [X2: real,A3: real,B3: real] :
      ( ( ord_less_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ X2 @ B3 ) )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ A3 @ B3 ) ) ) ).

% powr_less_cancel
thf(fact_8331_powr__less__mono,axiom,
    ! [A3: real,B3: real,X2: real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( ord_less_real @ one_one_real @ X2 )
       => ( ord_less_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).

% powr_less_mono
thf(fact_8332_replicate__length__same,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replic3825545231534752113n_assn @ ( size_s6829681357464350627n_assn @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8333_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X2: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8334_replicate__length__same,axiom,
    ! [Xs2: list_real,X2: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_real @ ( size_size_list_real @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8335_replicate__length__same,axiom,
    ! [Xs2: list_o,X2: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8336_replicate__length__same,axiom,
    ! [Xs2: list_nat,X2: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8337_replicate__length__same,axiom,
    ! [Xs2: list_int,X2: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( X3 = X2 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X2 )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_8338_replicate__eqI,axiom,
    ! [Xs2: list_P8527749157015355191n_assn,N2: nat,X2: produc6575502325842934193n_assn] :
      ( ( ( size_s6829681357464350627n_assn @ Xs2 )
        = N2 )
     => ( ! [Y3: produc6575502325842934193n_assn] :
            ( ( member7957490590177025114n_assn @ Y3 @ ( set_Pr1139785259514867910n_assn @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replic3825545231534752113n_assn @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8339_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N2: nat,X2: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8340_replicate__eqI,axiom,
    ! [Xs2: list_real,N2: nat,X2: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replicate_real @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8341_replicate__eqI,axiom,
    ! [Xs2: list_o,N2: nat,X2: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N2 )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replicate_o @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8342_replicate__eqI,axiom,
    ! [Xs2: list_nat,N2: nat,X2: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replicate_nat @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8343_replicate__eqI,axiom,
    ! [Xs2: list_int,N2: nat,X2: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X2 ) )
       => ( Xs2
          = ( replicate_int @ N2 @ X2 ) ) ) ) ).

% replicate_eqI
thf(fact_8344_map__replicate__const,axiom,
    ! [K: nat,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_nat
        @ ^ [X: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_nat @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8345_map__replicate__const,axiom,
    ! [K: real,Lst: list_VEBT_VEBT] :
      ( ( map_VEBT_VEBT_real
        @ ^ [X: vEBT_VEBT] : K
        @ Lst )
      = ( replicate_real @ ( size_s6755466524823107622T_VEBT @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8346_map__replicate__const,axiom,
    ! [K: assn,Lst: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn
        @ ^ [X: produc6575502325842934193n_assn] : K
        @ Lst )
      = ( replicate_assn @ ( size_s6829681357464350627n_assn @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8347_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_real] :
      ( ( map_real_VEBT_VEBT
        @ ^ [X: real] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_real @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8348_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_o] :
      ( ( map_o_VEBT_VEBT
        @ ^ [X: $o] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_o @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8349_map__replicate__const,axiom,
    ! [K: nat,Lst: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8350_map__replicate__const,axiom,
    ! [K: $o,Lst: list_nat] :
      ( ( map_nat_o
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_o @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8351_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_nat] :
      ( ( map_nat_VEBT_VEBT
        @ ^ [X: nat] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_nat @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8352_map__replicate__const,axiom,
    ! [K: vEBT_VEBT,Lst: list_int] :
      ( ( map_int_VEBT_VEBT
        @ ^ [X: int] : K
        @ Lst )
      = ( replicate_VEBT_VEBT @ ( size_size_list_int @ Lst ) @ K ) ) ).

% map_replicate_const
thf(fact_8353_powr__less__mono2,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ Y2 )
         => ( ord_less_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).

% powr_less_mono2
thf(fact_8354_powr__mono2_H,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ A3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ Y2 )
         => ( ord_less_eq_real @ ( powr_real @ Y2 @ A3 ) @ ( powr_real @ X2 @ A3 ) ) ) ) ) ).

% powr_mono2'
thf(fact_8355_gr__one__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ Y2 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y2 ) ) ) ) ).

% gr_one_powr
thf(fact_8356_powr__inj,axiom,
    ! [A3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ( powr_real @ A3 @ X2 )
            = ( powr_real @ A3 @ Y2 ) )
          = ( X2 = Y2 ) ) ) ) ).

% powr_inj
thf(fact_8357_ge__one__powr__ge__zero,axiom,
    ! [X2: real,A3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A3 )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A3 ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_8358_powr__mono__both,axiom,
    ! [A3: real,B3: real,X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ A3 @ B3 )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ Y2 )
           => ( ord_less_eq_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ Y2 @ B3 ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_8359_powr__le1,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X2 @ A3 ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_8360_powr__divide,axiom,
    ! [X2: real,Y2: real,A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( divide_divide_real @ X2 @ Y2 ) @ A3 )
          = ( divide_divide_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).

% powr_divide
thf(fact_8361_powr__mult,axiom,
    ! [X2: real,Y2: real,A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( powr_real @ ( times_times_real @ X2 @ Y2 ) @ A3 )
          = ( times_times_real @ ( powr_real @ X2 @ A3 ) @ ( powr_real @ Y2 @ A3 ) ) ) ) ) ).

% powr_mult
thf(fact_8362_log__base__powr,axiom,
    ! [A3: real,B3: real,X2: real] :
      ( ( A3 != zero_zero_real )
     => ( ( log @ ( powr_real @ A3 @ B3 ) @ X2 )
        = ( divide_divide_real @ ( log @ A3 @ X2 ) @ B3 ) ) ) ).

% log_base_powr
thf(fact_8363_log__powr,axiom,
    ! [X2: real,B3: real,Y2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( log @ B3 @ ( powr_real @ X2 @ Y2 ) )
        = ( times_times_real @ Y2 @ ( log @ B3 @ X2 ) ) ) ) ).

% log_powr
thf(fact_8364_ln__powr,axiom,
    ! [X2: real,Y2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X2 @ Y2 ) )
        = ( times_times_real @ Y2 @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_powr
thf(fact_8365_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_8366_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: produc6575502325842934193n_assn] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ X2 ) )
          = bot_bo1176836662018730877n_assn ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_Pr1139785259514867910n_assn @ ( replic3825545231534752113n_assn @ N2 @ X2 ) )
          = ( insert5290817439147925377n_assn @ X2 @ bot_bo1176836662018730877n_assn ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8367_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: vEBT_VEBT] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X2 ) )
          = ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8368_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X2 ) )
          = bot_bot_set_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X2 ) )
          = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8369_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X2 ) )
          = bot_bot_set_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X2 ) )
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8370_set__replicate__conv__if,axiom,
    ! [N2: nat,X2: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X2 ) )
          = bot_bot_set_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X2 ) )
          = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_8371_powr__realpow,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X2 @ N2 ) ) ) ).

% powr_realpow
thf(fact_8372_powr__less__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( powr_real @ B3 @ Y2 ) @ X2 )
          = ( ord_less_real @ Y2 @ ( log @ B3 @ X2 ) ) ) ) ) ).

% powr_less_iff
thf(fact_8373_less__powr__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y2 ) )
          = ( ord_less_real @ ( log @ B3 @ X2 ) @ Y2 ) ) ) ) ).

% less_powr_iff
thf(fact_8374_log__less__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ ( log @ B3 @ X2 ) @ Y2 )
          = ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y2 ) ) ) ) ) ).

% log_less_iff
thf(fact_8375_less__log__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_real @ Y2 @ ( log @ B3 @ X2 ) )
          = ( ord_less_real @ ( powr_real @ B3 @ Y2 ) @ X2 ) ) ) ) ).

% less_log_iff
thf(fact_8376_powr__minus__divide,axiom,
    ! [X2: real,A3: real] :
      ( ( powr_real @ X2 @ ( uminus_uminus_real @ A3 ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X2 @ A3 ) ) ) ).

% powr_minus_divide
thf(fact_8377_powr__neg__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).

% powr_neg_one
thf(fact_8378_powr__mult__base,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y2 ) )
        = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y2 ) ) ) ) ).

% powr_mult_base
thf(fact_8379_powr__le__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( powr_real @ B3 @ Y2 ) @ X2 )
          = ( ord_less_eq_real @ Y2 @ ( log @ B3 @ X2 ) ) ) ) ) ).

% powr_le_iff
thf(fact_8380_le__powr__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y2 ) )
          = ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y2 ) ) ) ) ).

% le_powr_iff
thf(fact_8381_log__le__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y2 )
          = ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y2 ) ) ) ) ) ).

% log_le_iff
thf(fact_8382_le__log__iff,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ one_one_real @ B3 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( ord_less_eq_real @ Y2 @ ( log @ B3 @ X2 ) )
          = ( ord_less_eq_real @ ( powr_real @ B3 @ Y2 ) @ X2 ) ) ) ) ).

% le_log_iff
thf(fact_8383_ln__powr__bound,axiom,
    ! [X2: real,A3: real] :
      ( ( ord_less_eq_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A3 ) @ A3 ) ) ) ) ).

% ln_powr_bound
thf(fact_8384_ln__powr__bound2,axiom,
    ! [X2: real,A3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A3 ) @ ( times_times_real @ ( powr_real @ A3 @ A3 ) @ X2 ) ) ) ) ).

% ln_powr_bound2
thf(fact_8385_add__log__eq__powr,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ Y2 @ ( log @ B3 @ X2 ) )
            = ( log @ B3 @ ( times_times_real @ ( powr_real @ B3 @ Y2 ) @ X2 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8386_log__add__eq__powr,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( plus_plus_real @ ( log @ B3 @ X2 ) @ Y2 )
            = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ Y2 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8387_minus__log__eq__powr,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ Y2 @ ( log @ B3 @ X2 ) )
            = ( log @ B3 @ ( divide_divide_real @ ( powr_real @ B3 @ Y2 ) @ X2 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8388_log__minus__eq__powr,axiom,
    ! [B3: real,X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ B3 )
     => ( ( B3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( minus_minus_real @ ( log @ B3 @ X2 ) @ Y2 )
            = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ ( uminus_uminus_real @ Y2 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8389_powr__half__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X2 ) ) ) ).

% powr_half_sqrt
thf(fact_8390_powr__neg__numeral,axiom,
    ! [X2: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8391_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( 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 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
          = ( 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.simps(3)
thf(fact_8392_vebt__buildup_Opelims,axiom,
    ! [X2: nat,Y2: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X2 )
        = Y2 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
       => ( ( ( X2 = zero_zero_nat )
           => ( ( Y2
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X2
                = ( suc @ zero_zero_nat ) )
             => ( ( Y2
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X2
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y2
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_8393_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8394_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8395_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8396_sum__gp,axiom,
    ! [N2: nat,M: nat,X2: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X2 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X2 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_8397_sin__arccos__abs,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y2 ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_8398_summable__complex__of__real,axiom,
    ! [F: nat > real] :
      ( ( summable_complex
        @ ^ [N: nat] : ( real_V4546457046886955230omplex @ ( F @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_complex_of_real
thf(fact_8399_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu: nat] : zero_zero_nat
        @ A4 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_8400_sum_Oneutral__const,axiom,
    ! [A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu: complex] : zero_zero_complex
        @ A4 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_8401_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu: nat] : zero_zero_real
        @ A4 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_8402_sum_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu: int] : zero_zero_int
        @ A4 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_8403_of__nat__sum,axiom,
    ! [F: complex > nat,A4: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8404_of__nat__sum,axiom,
    ! [F: int > nat,A4: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8405_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8406_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri4939895301339042750nteger @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups7501900531339628137nteger
        @ ^ [X: nat] : ( semiri4939895301339042750nteger @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8407_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8408_of__nat__sum,axiom,
    ! [F: nat > nat,A4: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A4 ) ) ).

% of_nat_sum
thf(fact_8409_of__int__sum,axiom,
    ! [F: complex > int,A4: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( ring_17405671764205052669omplex @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8410_of__int__sum,axiom,
    ! [F: nat > int,A4: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8411_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups8778361861064173332t_real
        @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8412_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups3906332499630173760nt_rat
        @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8413_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_17408606157368542149l_num1 @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups8925579862173457374l_num1
        @ ^ [X: int] : ( ring_17408606157368542149l_num1 @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8414_of__int__sum,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A4 ) ) ).

% of_int_sum
thf(fact_8415_abs__sum__abs,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ( abs_abs_real
        @ ( groups6591440286371151544t_real
          @ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
          @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [A2: nat] : ( abs_abs_real @ ( F @ A2 ) )
        @ A4 ) ) ).

% abs_sum_abs
thf(fact_8416_abs__sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( abs_abs_int
        @ ( groups4538972089207619220nt_int
          @ ^ [A2: int] : ( abs_abs_int @ ( F @ A2 ) )
          @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [A2: int] : ( abs_abs_int @ ( F @ A2 ) )
        @ A4 ) ) ).

% abs_sum_abs
thf(fact_8417_of__real__sum,axiom,
    ! [F: complex > real,S: set_complex] :
      ( ( real_V4546457046886955230omplex @ ( groups5808333547571424918x_real @ F @ S ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( real_V4546457046886955230omplex @ ( F @ X ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8418_of__real__sum,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( real_V4546457046886955230omplex @ ( groups6591440286371151544t_real @ F @ S ) )
      = ( groups2073611262835488442omplex
        @ ^ [X: nat] : ( real_V4546457046886955230omplex @ ( F @ X ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8419_of__real__sum,axiom,
    ! [F: nat > real,S: set_nat] :
      ( ( real_V1803761363581548252l_real @ ( groups6591440286371151544t_real @ F @ S ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( real_V1803761363581548252l_real @ ( F @ X ) )
        @ S ) ) ).

% of_real_sum
thf(fact_8420_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_8421_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8422_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8423_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_8424_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8425_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_8426_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_8427_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_8428_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8429_sum_Oempty,axiom,
    ! [G: nat > nat] :
      ( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_8430_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8431_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8432_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: code_integer > real] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_8433_sum_Oinfinite,axiom,
    ! [A4: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8434_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8435_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8436_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_8437_sum_Oinfinite,axiom,
    ! [A4: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G @ A4 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_8438_sum_Oinfinite,axiom,
    ! [A4: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G @ A4 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_8439_sum_Oinfinite,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat] :
      ( ~ ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G @ A4 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_8440_sum__eq__0__iff,axiom,
    ! [F6: set_int,F: int > nat] :
      ( ( finite_finite_int @ F6 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F6 )
          = zero_zero_nat )
        = ( ! [X: int] :
              ( ( member_int @ X @ F6 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8441_sum__eq__0__iff,axiom,
    ! [F6: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F6 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F6 )
          = zero_zero_nat )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ F6 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8442_sum__eq__0__iff,axiom,
    ! [F6: set_Code_integer,F: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ F6 )
     => ( ( ( groups7237345082560585321er_nat @ F @ F6 )
          = zero_zero_nat )
        = ( ! [X: code_integer] :
              ( ( member_Code_integer @ X @ F6 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8443_sum__eq__0__iff,axiom,
    ! [F6: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F6 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F6 )
          = zero_zero_nat )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ F6 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_8444_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_8445_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_8446_of__real__eq__0__iff,axiom,
    ! [X2: real] :
      ( ( ( real_V1803761363581548252l_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8447_of__real__eq__0__iff,axiom,
    ! [X2: real] :
      ( ( ( real_V4546457046886955230omplex @ X2 )
        = zero_zero_complex )
      = ( X2 = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8448_of__real__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( real_V1803761363581548252l_real @ X2 )
        = one_one_real )
      = ( X2 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_8449_of__real__eq__1__iff,axiom,
    ! [X2: real] :
      ( ( ( real_V4546457046886955230omplex @ X2 )
        = one_one_complex )
      = ( X2 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_8450_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_8451_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_8452_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_8453_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_8454_of__real__divide,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y2 ) ) ) ).

% of_real_divide
thf(fact_8455_of__real__divide,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y2 ) )
      = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y2 ) ) ) ).

% of_real_divide
thf(fact_8456_of__real__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X2 @ N2 ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X2 ) @ N2 ) ) ).

% of_real_power
thf(fact_8457_of__real__power,axiom,
    ! [X2: real,N2: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X2 @ N2 ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X2 ) @ N2 ) ) ).

% of_real_power
thf(fact_8458_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_8459_sum_Odelta_H,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8460_sum_Odelta_H,axiom,
    ! [S4: set_real,A3: real,B3: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( ( member_real @ A3 @ S4 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_real @ A3 @ S4 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8461_sum_Odelta_H,axiom,
    ! [S4: set_int,A3: int,B3: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( ( member_int @ A3 @ S4 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_int @ A3 @ S4 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8462_sum_Odelta_H,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8463_sum_Odelta_H,axiom,
    ! [S4: set_Code_integer,A3: code_integer,B3: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_8464_sum_Odelta_H,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8465_sum_Odelta_H,axiom,
    ! [S4: set_real,A3: real,B3: real > rat] :
      ( ( finite_finite_real @ S4 )
     => ( ( ( member_real @ A3 @ S4 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_real @ A3 @ S4 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8466_sum_Odelta_H,axiom,
    ! [S4: set_nat,A3: nat,B3: nat > rat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( ( member_nat @ A3 @ S4 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_nat @ A3 @ S4 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8467_sum_Odelta_H,axiom,
    ! [S4: set_int,A3: int,B3: int > rat] :
      ( ( finite_finite_int @ S4 )
     => ( ( ( member_int @ A3 @ S4 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_int @ A3 @ S4 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8468_sum_Odelta_H,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > rat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A3 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_8469_sum_Odelta,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8470_sum_Odelta,axiom,
    ! [S4: set_real,A3: real,B3: real > real] :
      ( ( finite_finite_real @ S4 )
     => ( ( ( member_real @ A3 @ S4 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_real @ A3 @ S4 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8471_sum_Odelta,axiom,
    ! [S4: set_int,A3: int,B3: int > real] :
      ( ( finite_finite_int @ S4 )
     => ( ( ( member_int @ A3 @ S4 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_int @ A3 @ S4 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8472_sum_Odelta,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8473_sum_Odelta,axiom,
    ! [S4: set_Code_integer,A3: code_integer,B3: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_real )
              @ S4 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_8474_sum_Odelta,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8475_sum_Odelta,axiom,
    ! [S4: set_real,A3: real,B3: real > rat] :
      ( ( finite_finite_real @ S4 )
     => ( ( ( member_real @ A3 @ S4 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_real @ A3 @ S4 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8476_sum_Odelta,axiom,
    ! [S4: set_nat,A3: nat,B3: nat > rat] :
      ( ( finite_finite_nat @ S4 )
     => ( ( ( member_nat @ A3 @ S4 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_nat @ A3 @ S4 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8477_sum_Odelta,axiom,
    ! [S4: set_int,A3: int,B3: int > rat] :
      ( ( finite_finite_int @ S4 )
     => ( ( ( member_int @ A3 @ S4 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_int @ A3 @ S4 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8478_sum_Odelta,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > rat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = ( B3 @ A3 ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
              @ S4 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_8479_sum__abs,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_8480_sum__abs,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_8481_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_8482_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_8483_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_8484_sum__abs__ge__zero,axiom,
    ! [F: int > int,A4: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_8485_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8486_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8487_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_8488_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_8489_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > word_N3645301735248828278l_num1] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2996710295995929986l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_z3563351764282998399l_num1 ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2996710295995929986l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_p361126936061061375l_num1 @ ( groups2996710295995929986l_num1 @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8490_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8491_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8492_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8493_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_8494_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_8495_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_8496_sum__zero__power,axiom,
    ! [A4: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A4 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_8497_norm__of__real__add1,axiom,
    ! [X2: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8498_norm__of__real__add1,axiom,
    ! [X2: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X2 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8499_norm__of__real__addn,axiom,
    ! [X2: real,B3: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( numeral_numeral_real @ B3 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B3 ) ) ) ) ).

% norm_of_real_addn
thf(fact_8500_norm__of__real__addn,axiom,
    ! [X2: real,B3: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X2 ) @ ( numera6690914467698888265omplex @ B3 ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X2 @ ( numeral_numeral_real @ B3 ) ) ) ) ).

% norm_of_real_addn
thf(fact_8501_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_8502_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_8503_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_8504_sum__zero__power_H,axiom,
    ! [A4: set_nat,C: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_8505_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_8506_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_8507_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_8508_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_8509_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > real,A4: set_VEBT_VEBT] :
      ( ( ( groups2240296850493347238T_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8510_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A4: set_int] :
      ( ( ( groups8778361861064173332t_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8511_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A4: set_real] :
      ( ( ( groups8097168146408367636l_real @ G @ A4 )
       != zero_zero_real )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8512_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A4: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A: nat] :
            ( ( member_nat @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8513_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > rat,A4: set_VEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8514_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A4: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8515_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A4: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A4 )
       != zero_zero_rat )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8516_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: vEBT_VEBT > nat,A4: set_VEBT_VEBT] :
      ( ( ( groups771621172384141258BT_nat @ G @ A4 )
       != zero_zero_nat )
     => ~ ! [A: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8517_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > nat,A4: set_int] :
      ( ( ( groups4541462559716669496nt_nat @ G @ A4 )
       != zero_zero_nat )
     => ~ ! [A: int] :
            ( ( member_int @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8518_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > nat,A4: set_real] :
      ( ( ( groups1935376822645274424al_nat @ G @ A4 )
       != zero_zero_nat )
     => ~ ! [A: real] :
            ( ( member_real @ A @ A4 )
           => ( ( G @ A )
              = zero_zero_nat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_8519_sum_Oneutral,axiom,
    ! [A4: set_nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A4 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_8520_sum_Oneutral,axiom,
    ! [A4: set_complex,G: complex > complex] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A4 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_8521_sum_Oneutral,axiom,
    ! [A4: set_nat,G: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A4 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_8522_sum_Oneutral,axiom,
    ! [A4: set_int,G: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( G @ X3 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G @ A4 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_8523_sum_Oswap,axiom,
    ! [G: nat > nat > nat,B6: set_nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G @ I4 ) @ B6 )
        @ A4 )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A4 )
        @ B6 ) ) ).

% sum.swap
thf(fact_8524_sum_Oswap,axiom,
    ! [G: complex > complex > complex,B6: set_complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [I4: complex] : ( groups7754918857620584856omplex @ ( G @ I4 ) @ B6 )
        @ A4 )
      = ( groups7754918857620584856omplex
        @ ^ [J3: complex] :
            ( groups7754918857620584856omplex
            @ ^ [I4: complex] : ( G @ I4 @ J3 )
            @ A4 )
        @ B6 ) ) ).

% sum.swap
thf(fact_8525_sum_Oswap,axiom,
    ! [G: nat > nat > real,B6: set_nat,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( G @ I4 ) @ B6 )
        @ A4 )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ I4 @ J3 )
            @ A4 )
        @ B6 ) ) ).

% sum.swap
thf(fact_8526_sum_Oswap,axiom,
    ! [G: int > int > int,B6: set_int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [I4: int] : ( groups4538972089207619220nt_int @ ( G @ I4 ) @ B6 )
        @ A4 )
      = ( groups4538972089207619220nt_int
        @ ^ [J3: int] :
            ( groups4538972089207619220nt_int
            @ ^ [I4: int] : ( G @ I4 @ J3 )
            @ A4 )
        @ B6 ) ) ).

% sum.swap
thf(fact_8527_norm__sum,axiom,
    ! [F: nat > complex,A4: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_8528_norm__sum,axiom,
    ! [F: complex > complex,A4: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A4 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_8529_norm__sum,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
        @ A4 ) ) ).

% norm_sum
thf(fact_8530_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8531_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ K5 ) @ ( groups136491112297645522BT_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8532_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8533_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8534_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > nat,G: vEBT_VEBT > nat] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ K5 ) @ ( groups771621172384141258BT_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8535_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8536_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8537_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8538_sum__mono,axiom,
    ! [K5: set_VEBT_VEBT,F: vEBT_VEBT > int,G: vEBT_VEBT > int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups769130701875090982BT_int @ F @ K5 ) @ ( groups769130701875090982BT_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8539_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ K5 )
         => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_8540_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_8541_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_8542_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_8543_sum_Odistrib,axiom,
    ! [G: int > int,H2: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H2 @ X ) )
        @ A4 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H2 @ A4 ) ) ) ).

% sum.distrib
thf(fact_8544_sum__product,axiom,
    ! [F: nat > nat,A4: set_nat,G: nat > nat,B6: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B6 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B6 )
        @ A4 ) ) ).

% sum_product
thf(fact_8545_sum__product,axiom,
    ! [F: complex > complex,A4: set_complex,G: complex > complex,B6: set_complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ B6 ) )
      = ( groups7754918857620584856omplex
        @ ^ [I4: complex] :
            ( groups7754918857620584856omplex
            @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B6 )
        @ A4 ) ) ).

% sum_product
thf(fact_8546_sum__product,axiom,
    ! [F: nat > real,A4: set_nat,G: nat > real,B6: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ B6 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B6 )
        @ A4 ) ) ).

% sum_product
thf(fact_8547_sum__product,axiom,
    ! [F: int > int,A4: set_int,G: int > int,B6: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B6 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I4: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
            @ B6 )
        @ A4 ) ) ).

% sum_product
thf(fact_8548_sum__distrib__right,axiom,
    ! [F: nat > nat,A4: set_nat,R: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ R )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_8549_sum__distrib__right,axiom,
    ! [F: complex > complex,A4: set_complex,R: complex] :
      ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_8550_sum__distrib__right,axiom,
    ! [F: nat > real,A4: set_nat,R: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_8551_sum__distrib__right,axiom,
    ! [F: int > int,A4: set_int,R: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ R )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_8552_sum__distrib__left,axiom,
    ! [R: nat,F: nat > nat,A4: set_nat] :
      ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_8553_sum__distrib__left,axiom,
    ! [R: complex,F: complex > complex,A4: set_complex] :
      ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A4 ) )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( times_times_complex @ R @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_8554_sum__distrib__left,axiom,
    ! [R: real,F: nat > real,A4: set_nat] :
      ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( times_times_real @ R @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_8555_sum__distrib__left,axiom,
    ! [R: int,F: int > int,A4: set_int] :
      ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N: int] : ( times_times_int @ R @ ( F @ N ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_8556_sum__subtractf,axiom,
    ! [F: complex > complex,G: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( minus_minus_complex @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_8557_sum__subtractf,axiom,
    ! [F: nat > real,G: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_8558_sum__subtractf,axiom,
    ! [F: int > int,G: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( minus_minus_int @ ( F @ X ) @ ( G @ X ) )
        @ A4 )
      = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ).

% sum_subtractf
thf(fact_8559_sum__divide__distrib,axiom,
    ! [F: complex > complex,A4: set_complex,R: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A4 ) @ R )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_8560_sum__divide__distrib,axiom,
    ! [F: nat > real,A4: set_nat,R: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ R )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R )
        @ A4 ) ) ).

% sum_divide_distrib
thf(fact_8561_sum__negf,axiom,
    ! [F: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( uminus1482373934393186551omplex @ ( F @ X ) )
        @ A4 )
      = ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A4 ) ) ) ).

% sum_negf
thf(fact_8562_sum__negf,axiom,
    ! [F: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( uminus_uminus_real @ ( F @ X ) )
        @ A4 )
      = ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A4 ) ) ) ).

% sum_negf
thf(fact_8563_sum__negf,axiom,
    ! [F: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( uminus_uminus_int @ ( F @ X ) )
        @ A4 )
      = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) ) ) ).

% sum_negf
thf(fact_8564_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_nat,G: vEBT_VEBT > nat > nat,R4: vEBT_VEBT > nat > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( groups771621172384141258BT_nat
            @ ^ [X: vEBT_VEBT] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups771621172384141258BT_nat
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8565_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B6: set_nat,G: real > nat > nat,R4: real > nat > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X: real] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8566_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B6: set_nat,G: int > nat > nat,R4: int > nat > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X: int] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8567_sum_Oswap__restrict,axiom,
    ! [A4: set_complex,B6: set_nat,G: complex > nat > nat,R4: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8568_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B6: set_nat,G: code_integer > nat > nat,R4: code_integer > nat > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite_finite_nat @ B6 )
       => ( ( groups7237345082560585321er_nat
            @ ^ [X: code_integer] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups7237345082560585321er_nat
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8569_sum_Oswap__restrict,axiom,
    ! [A4: set_VEBT_VEBT,B6: set_complex,G: vEBT_VEBT > complex > complex,R4: vEBT_VEBT > complex > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ( groups1794756597179926696omplex
            @ ^ [X: vEBT_VEBT] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups1794756597179926696omplex
                @ ^ [X: vEBT_VEBT] : ( G @ X @ Y )
                @ ( collect_VEBT_VEBT
                  @ ^ [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8570_sum_Oswap__restrict,axiom,
    ! [A4: set_real,B6: set_complex,G: real > complex > complex,R4: real > complex > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X: real] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8571_sum_Oswap__restrict,axiom,
    ! [A4: set_nat,B6: set_complex,G: nat > complex > complex,R4: nat > complex > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X: nat] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X: nat] : ( G @ X @ Y )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8572_sum_Oswap__restrict,axiom,
    ! [A4: set_int,B6: set_complex,G: int > complex > complex,R4: int > complex > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X: int] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8573_sum_Oswap__restrict,axiom,
    ! [A4: set_Code_integer,B6: set_complex,G: code_integer > complex > complex,R4: code_integer > complex > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( finite3207457112153483333omplex @ B6 )
       => ( ( groups8024822376189712711omplex
            @ ^ [X: code_integer] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B6 )
                      & ( R4 @ X @ Y ) ) ) )
            @ A4 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups8024822376189712711omplex
                @ ^ [X: code_integer] : ( G @ X @ Y )
                @ ( collect_Code_integer
                  @ ^ [X: code_integer] :
                      ( ( member_Code_integer @ X @ A4 )
                      & ( R4 @ X @ Y ) ) ) )
            @ B6 ) ) ) ) ).

% sum.swap_restrict
thf(fact_8574_mod__sum__eq,axiom,
    ! [F: nat > nat,A3: nat,A4: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A3 )
          @ A4 )
        @ A3 )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ A3 ) ) ).

% mod_sum_eq
thf(fact_8575_mod__sum__eq,axiom,
    ! [F: int > int,A3: int,A4: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A3 )
          @ A4 )
        @ A3 )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ A3 ) ) ).

% mod_sum_eq
thf(fact_8576_summable__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat > real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N: nat] :
            ( groups2240296850493347238T_real
            @ ^ [I4: vEBT_VEBT] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8577_summable__sum,axiom,
    ! [I5: set_int,F: int > nat > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N: nat] :
            ( groups8778361861064173332t_real
            @ ^ [I4: int] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8578_summable__sum,axiom,
    ! [I5: set_real,F: real > nat > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N: nat] :
            ( groups8097168146408367636l_real
            @ ^ [I4: real] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8579_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N: nat] :
            ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8580_summable__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat > complex] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N: nat] :
            ( groups1794756597179926696omplex
            @ ^ [I4: vEBT_VEBT] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8581_summable__sum,axiom,
    ! [I5: set_int,F: int > nat > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N: nat] :
            ( groups3049146728041665814omplex
            @ ^ [I4: int] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8582_summable__sum,axiom,
    ! [I5: set_real,F: real > nat > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N: nat] :
            ( groups5754745047067104278omplex
            @ ^ [I4: real] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8583_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_nat @ ( F @ I2 ) ) )
     => ( summable_nat
        @ ^ [N: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8584_summable__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( summable_complex
        @ ^ [N: nat] :
            ( groups7754918857620584856omplex
            @ ^ [I4: complex] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8585_summable__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( summable_real
        @ ^ [N: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( F @ I4 @ N )
            @ I5 ) ) ) ).

% summable_sum
thf(fact_8586_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8587_sum__nonpos,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8588_sum__nonpos,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_8589_sum__nonpos,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8590_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8591_sum__nonpos,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8592_sum__nonpos,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_8593_sum__nonpos,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8594_sum__nonpos,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8595_sum__nonpos,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_8596_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8597_sum__nonneg,axiom,
    ! [A4: set_int,F: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8598_sum__nonneg,axiom,
    ! [A4: set_real,F: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8599_sum__nonneg,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8600_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8601_sum__nonneg,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8602_sum__nonneg,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8603_sum__nonneg,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups771621172384141258BT_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8604_sum__nonneg,axiom,
    ! [A4: set_int,F: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8605_sum__nonneg,axiom,
    ! [A4: set_real,F: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_8606_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > rat,I5: set_VEBT_VEBT,G: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( ( groups136491112297645522BT_rat @ F @ I5 )
        = ( groups136491112297645522BT_rat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8607_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8608_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8609_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8610_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8611_sum__mono__inv,axiom,
    ! [F: code_integer > rat,I5: set_Code_integer,G: code_integer > rat,I: code_integer] :
      ( ( ( groups6602215022474089585er_rat @ F @ I5 )
        = ( groups6602215022474089585er_rat @ G @ I5 ) )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_Code_integer @ I @ I5 )
         => ( ( finite6017078050557962740nteger @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8612_sum__mono__inv,axiom,
    ! [F: vEBT_VEBT > nat,I5: set_VEBT_VEBT,G: vEBT_VEBT > nat,I: vEBT_VEBT] :
      ( ( ( groups771621172384141258BT_nat @ F @ I5 )
        = ( groups771621172384141258BT_nat @ G @ I5 ) )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_VEBT_VEBT @ I @ I5 )
         => ( ( finite5795047828879050333T_VEBT @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8613_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8614_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8615_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_8616_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8617_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F @ ( suc @ X3 ) )
              = ( G @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A4 )
          = ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_8618_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > real,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups2240296850493347238T_real @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups2240296850493347238T_real
          @ ^ [X: vEBT_VEBT] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8619_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8620_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8621_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8622_sum_Ointer__filter,axiom,
    ! [A4: set_Code_integer,G: code_integer > real,P: code_integer > $o] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( collect_Code_integer
            @ ^ [X: code_integer] :
                ( ( member_Code_integer @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups1270011288395367621r_real
          @ ^ [X: code_integer] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8623_sum_Ointer__filter,axiom,
    ! [A4: set_VEBT_VEBT,G: vEBT_VEBT > rat,P: vEBT_VEBT > $o] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( groups136491112297645522BT_rat @ G
          @ ( collect_VEBT_VEBT
            @ ^ [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups136491112297645522BT_rat
          @ ^ [X: vEBT_VEBT] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8624_sum_Ointer__filter,axiom,
    ! [A4: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8625_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8626_sum_Ointer__filter,axiom,
    ! [A4: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8627_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A4 )
                & ( P @ X ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_8628_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8629_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_8630_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8631_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_8632_suminf__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat > real] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] :
              ( groups2240296850493347238T_real
              @ ^ [I4: vEBT_VEBT] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups2240296850493347238T_real
          @ ^ [I4: vEBT_VEBT] : ( suminf_real @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8633_suminf__sum,axiom,
    ! [I5: set_int,F: int > nat > real] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] :
              ( groups8778361861064173332t_real
              @ ^ [I4: int] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups8778361861064173332t_real
          @ ^ [I4: int] : ( suminf_real @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8634_suminf__sum,axiom,
    ! [I5: set_real,F: real > nat > real] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] :
              ( groups8097168146408367636l_real
              @ ^ [I4: real] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups8097168146408367636l_real
          @ ^ [I4: real] : ( suminf_real @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8635_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > complex] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] :
              ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( suminf_complex @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8636_suminf__sum,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > nat > complex] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] :
              ( groups1794756597179926696omplex
              @ ^ [I4: vEBT_VEBT] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups1794756597179926696omplex
          @ ^ [I4: vEBT_VEBT] : ( suminf_complex @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8637_suminf__sum,axiom,
    ! [I5: set_int,F: int > nat > complex] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] :
              ( groups3049146728041665814omplex
              @ ^ [I4: int] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups3049146728041665814omplex
          @ ^ [I4: int] : ( suminf_complex @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8638_suminf__sum,axiom,
    ! [I5: set_real,F: real > nat > complex] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] :
              ( groups5754745047067104278omplex
              @ ^ [I4: real] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups5754745047067104278omplex
          @ ^ [I4: real] : ( suminf_complex @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8639_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > nat] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_nat @ ( F @ I2 ) ) )
     => ( ( suminf_nat
          @ ^ [N: nat] :
              ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [I4: nat] : ( suminf_nat @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8640_suminf__sum,axiom,
    ! [I5: set_complex,F: complex > nat > complex] :
      ( ! [I2: complex] :
          ( ( member_complex @ I2 @ I5 )
         => ( summable_complex @ ( F @ I2 ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] :
              ( groups7754918857620584856omplex
              @ ^ [I4: complex] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups7754918857620584856omplex
          @ ^ [I4: complex] : ( suminf_complex @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8641_suminf__sum,axiom,
    ! [I5: set_nat,F: nat > nat > real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( summable_real @ ( F @ I2 ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( F @ I4 @ N )
              @ I5 ) )
        = ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( suminf_real @ ( F @ I4 ) )
          @ I5 ) ) ) ).

% suminf_sum
thf(fact_8642_summable__of__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( summable_real
        @ ^ [N: nat] : ( real_V1803761363581548252l_real @ ( X9 @ N ) ) ) ) ).

% summable_of_real
thf(fact_8643_summable__of__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( summable_complex
        @ ^ [N: nat] : ( real_V4546457046886955230omplex @ ( X9 @ N ) ) ) ) ).

% summable_of_real
thf(fact_8644_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8645_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8646_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: int] :
                ( ( member_int @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8647_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8648_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X: code_integer] :
                ( ( member_Code_integer @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8649_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8650_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8651_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8652_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8653_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A4 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_8654_sum__le__included,axiom,
    ! [S: set_int,T2: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8655_sum__le__included,axiom,
    ! [S: set_int,T2: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T2 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa3: complex] :
                    ( ( member_complex @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8656_sum__le__included,axiom,
    ! [S: set_int,T2: set_Code_integer,G: code_integer > real,I: code_integer > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite6017078050557962740nteger @ T2 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8657_sum__le__included,axiom,
    ! [S: set_complex,T2: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8658_sum__le__included,axiom,
    ! [S: set_complex,T2: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T2 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa3: complex] :
                    ( ( member_complex @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8659_sum__le__included,axiom,
    ! [S: set_complex,T2: set_Code_integer,G: code_integer > real,I: code_integer > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite6017078050557962740nteger @ T2 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8660_sum__le__included,axiom,
    ! [S: set_Code_integer,T2: set_int,G: int > real,I: int > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite_finite_int @ T2 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa3: int] :
                    ( ( member_int @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8661_sum__le__included,axiom,
    ! [S: set_Code_integer,T2: set_complex,G: complex > real,I: complex > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite3207457112153483333omplex @ T2 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa3: complex] :
                    ( ( member_complex @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8662_sum__le__included,axiom,
    ! [S: set_Code_integer,T2: set_Code_integer,G: code_integer > real,I: code_integer > code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ( finite6017078050557962740nteger @ T2 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ T2 )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S )
               => ? [Xa3: code_integer] :
                    ( ( member_Code_integer @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ S ) @ ( groups1270011288395367621r_real @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8663_sum__le__included,axiom,
    ! [S: set_nat,T2: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T2 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T2 )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ T2 )
                    & ( ( I @ Xa3 )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa3 ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T2 ) ) ) ) ) ) ).

% sum_le_included
thf(fact_8664_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8665_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8666_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
              & ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8667_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8668_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8669_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8670_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
              & ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8671_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8672_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8673_sum__strict__mono__ex1,axiom,
    ! [A4: set_Code_integer,F: code_integer > nat,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
       => ( ? [X4: code_integer] :
              ( ( member_Code_integer @ X4 @ A4 )
              & ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups7237345082560585321er_nat @ F @ A4 ) @ ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_8674_sum_Orelated,axiom,
    ! [R4: real > real > $o,S4: set_int,H2: int > real,G: int > real] :
      ( ( R4 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X24 @ Y23 ) ) )
       => ( ( finite_finite_int @ S4 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups8778361861064173332t_real @ H2 @ S4 ) @ ( groups8778361861064173332t_real @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8675_sum_Orelated,axiom,
    ! [R4: real > real > $o,S4: set_complex,H2: complex > real,G: complex > real] :
      ( ( R4 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X24 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S4 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups5808333547571424918x_real @ H2 @ S4 ) @ ( groups5808333547571424918x_real @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8676_sum_Orelated,axiom,
    ! [R4: real > real > $o,S4: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( R4 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X24: real,Y23: real] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X24 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S4 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups1270011288395367621r_real @ H2 @ S4 ) @ ( groups1270011288395367621r_real @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8677_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S4: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S4 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups2906978787729119204at_rat @ H2 @ S4 ) @ ( groups2906978787729119204at_rat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8678_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S4: set_int,H2: int > rat,G: int > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y23 ) ) )
       => ( ( finite_finite_int @ S4 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups3906332499630173760nt_rat @ H2 @ S4 ) @ ( groups3906332499630173760nt_rat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8679_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S4: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S4 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups5058264527183730370ex_rat @ H2 @ S4 ) @ ( groups5058264527183730370ex_rat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8680_sum_Orelated,axiom,
    ! [R4: rat > rat > $o,S4: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
      ( ( R4 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X24: rat,Y23: rat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X24 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S4 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups6602215022474089585er_rat @ H2 @ S4 ) @ ( groups6602215022474089585er_rat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8681_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S4: set_int,H2: int > nat,G: int > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X24: nat,Y23: nat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X24 @ Y23 ) ) )
       => ( ( finite_finite_int @ S4 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups4541462559716669496nt_nat @ H2 @ S4 ) @ ( groups4541462559716669496nt_nat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8682_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S4: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X24: nat,Y23: nat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X24 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S4 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups5693394587270226106ex_nat @ H2 @ S4 ) @ ( groups5693394587270226106ex_nat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8683_sum_Orelated,axiom,
    ! [R4: nat > nat > $o,S4: set_Code_integer,H2: code_integer > nat,G: code_integer > nat] :
      ( ( R4 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X24: nat,Y23: nat] :
            ( ( ( R4 @ X15 @ X24 )
              & ( R4 @ Y15 @ Y23 ) )
           => ( R4 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X24 @ Y23 ) ) )
       => ( ( finite6017078050557962740nteger @ S4 )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
           => ( R4 @ ( groups7237345082560585321er_nat @ H2 @ S4 ) @ ( groups7237345082560585321er_nat @ G @ S4 ) ) ) ) ) ) ).

% sum.related
thf(fact_8684_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8685_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8686_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8687_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8688_sum__strict__mono,axiom,
    ! [A4: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ A4 )
             => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8689_sum__strict__mono,axiom,
    ! [A4: set_VEBT_VEBT,F: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( A4 != bot_bo8194388402131092736T_VEBT )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8690_sum__strict__mono,axiom,
    ! [A4: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( A4 != bot_bot_set_complex )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8691_sum__strict__mono,axiom,
    ! [A4: set_Code_integer,F: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( A4 != bot_bo3990330152332043303nteger )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8692_sum__strict__mono,axiom,
    ! [A4: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8693_sum__strict__mono,axiom,
    ! [A4: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ A4 )
             => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_8694_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_VEBT_VEBT,S4: set_VEBT_VEBT,I: vEBT_VEBT > vEBT_VEBT,J: vEBT_VEBT > vEBT_VEBT,T7: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A ) @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B ) @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) ) )
               => ( ! [A: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups2240296850493347238T_real @ G @ S4 )
                        = ( groups2240296850493347238T_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8695_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_real,S4: set_VEBT_VEBT,I: real > vEBT_VEBT,J: vEBT_VEBT > real,T7: set_real,G: vEBT_VEBT > real,H2: real > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
               => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T7 @ T6 ) ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: real] :
                    ( ( member_real @ B @ ( minus_minus_set_real @ T7 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B ) @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) ) )
               => ( ! [A: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( ( member_real @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups2240296850493347238T_real @ G @ S4 )
                        = ( groups8097168146408367636l_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8696_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_VEBT_VEBT,S4: set_real,I: vEBT_VEBT > real,J: real > vEBT_VEBT,T7: set_VEBT_VEBT,G: real > real,H2: vEBT_VEBT > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite5795047828879050333T_VEBT @ T6 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
               => ( member_VEBT_VEBT @ ( J @ A ) @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ T7 @ T6 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S4 @ S7 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S4 )
                        = ( groups2240296850493347238T_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8697_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_real,S4: set_real,I: real > real,J: real > real,T7: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_real @ T6 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
               => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T7 @ T6 ) ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: real] :
                    ( ( member_real @ B @ ( minus_minus_set_real @ T7 @ T6 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S4 @ S7 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: real] :
                        ( ( member_real @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S4 )
                        = ( groups8097168146408367636l_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8698_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_int,S4: set_VEBT_VEBT,I: int > vEBT_VEBT,J: vEBT_VEBT > int,T7: set_int,G: vEBT_VEBT > real,H2: int > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
               => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T7 @ T6 ) ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: int] :
                    ( ( member_int @ B @ ( minus_minus_set_int @ T7 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B ) @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) ) )
               => ( ! [A: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( ( member_int @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups2240296850493347238T_real @ G @ S4 )
                        = ( groups8778361861064173332t_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8699_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_int,S4: set_real,I: int > real,J: real > int,T7: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite_finite_int @ T6 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
               => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T7 @ T6 ) ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: int] :
                    ( ( member_int @ B @ ( minus_minus_set_int @ T7 @ T6 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S4 @ S7 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: int] :
                        ( ( member_int @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S4 )
                        = ( groups8778361861064173332t_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8700_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_complex,S4: set_VEBT_VEBT,I: complex > vEBT_VEBT,J: vEBT_VEBT > complex,T7: set_complex,G: vEBT_VEBT > real,H2: complex > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
               => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T7 @ T6 ) ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: complex] :
                    ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T7 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B ) @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) ) )
               => ( ! [A: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( ( member_complex @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups2240296850493347238T_real @ G @ S4 )
                        = ( groups5808333547571424918x_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8701_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_complex,S4: set_real,I: complex > real,J: real > complex,T7: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite3207457112153483333omplex @ T6 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
               => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T7 @ T6 ) ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: complex] :
                    ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T7 @ T6 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S4 @ S7 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: complex] :
                        ( ( member_complex @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S4 )
                        = ( groups5808333547571424918x_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8702_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_VEBT_VEBT,T6: set_Code_integer,S4: set_VEBT_VEBT,I: code_integer > vEBT_VEBT,J: vEBT_VEBT > code_integer,T7: set_Code_integer,G: vEBT_VEBT > real,H2: code_integer > real] :
      ( ( finite5795047828879050333T_VEBT @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) )
               => ( member_Code_integer @ ( J @ A ) @ ( minus_2355218937544613996nteger @ T7 @ T6 ) ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: code_integer] :
                    ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ T7 @ T6 ) )
                   => ( member_VEBT_VEBT @ ( I @ B ) @ ( minus_5127226145743854075T_VEBT @ S4 @ S7 ) ) )
               => ( ! [A: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: code_integer] :
                        ( ( member_Code_integer @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups2240296850493347238T_real @ G @ S4 )
                        = ( groups1270011288395367621r_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8703_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S7: set_real,T6: set_Code_integer,S4: set_real,I: code_integer > real,J: real > code_integer,T7: set_Code_integer,G: real > real,H2: code_integer > real] :
      ( ( finite_finite_real @ S7 )
     => ( ( finite6017078050557962740nteger @ T6 )
       => ( ! [A: real] :
              ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
             => ( ( I @ ( J @ A ) )
                = A ) )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ S4 @ S7 ) )
               => ( member_Code_integer @ ( J @ A ) @ ( minus_2355218937544613996nteger @ T7 @ T6 ) ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ T7 @ T6 ) )
                 => ( ( J @ ( I @ B ) )
                    = B ) )
             => ( ! [B: code_integer] :
                    ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ T7 @ T6 ) )
                   => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S4 @ S7 ) ) )
               => ( ! [A: real] :
                      ( ( member_real @ A @ S7 )
                     => ( ( G @ A )
                        = zero_zero_real ) )
                 => ( ! [B: code_integer] :
                        ( ( member_Code_integer @ B @ T6 )
                       => ( ( H2 @ B )
                          = zero_zero_real ) )
                   => ( ! [A: real] :
                          ( ( member_real @ A @ S4 )
                         => ( ( H2 @ ( J @ A ) )
                            = ( G @ A ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S4 )
                        = ( groups1270011288395367621r_real @ H2 @ T7 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_8704_nonzero__of__real__divide,axiom,
    ! [Y2: real,X2: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X2 @ Y2 ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X2 ) @ ( real_V1803761363581548252l_real @ Y2 ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_8705_nonzero__of__real__divide,axiom,
    ! [Y2: real,X2: real] :
      ( ( Y2 != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X2 @ Y2 ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X2 ) @ ( real_V4546457046886955230omplex @ Y2 ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_8706_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,B6: real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = B6 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8707_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B6: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B6 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8708_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B6: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B6 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8709_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B6: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B6 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8710_sum__nonneg__leq__bound,axiom,
    ! [S: set_Code_integer,F: code_integer > real,B6: real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = B6 )
         => ( ( member_Code_integer @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8711_sum__nonneg__leq__bound,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,B6: rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = B6 )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8712_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B6: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B6 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8713_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B6: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B6 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8714_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B6: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B6 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8715_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B6: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B6 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B6 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_8716_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > real,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups2240296850493347238T_real @ F @ S )
            = zero_zero_real )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8717_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8718_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8719_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8720_sum__nonneg__0,axiom,
    ! [S: set_Code_integer,F: code_integer > real,I: code_integer] :
      ( ( finite6017078050557962740nteger @ S )
     => ( ! [I2: code_integer] :
            ( ( member_Code_integer @ I2 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
       => ( ( ( groups1270011288395367621r_real @ F @ S )
            = zero_zero_real )
         => ( ( member_Code_integer @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8721_sum__nonneg__0,axiom,
    ! [S: set_VEBT_VEBT,F: vEBT_VEBT > rat,I: vEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ S )
     => ( ! [I2: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups136491112297645522BT_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_VEBT_VEBT @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8722_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I2: real] :
            ( ( member_real @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8723_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I2: nat] :
            ( ( member_nat @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8724_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I2: int] :
            ( ( member_int @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8725_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I2: complex] :
            ( ( member_complex @ I2 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_8726_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8727_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8728_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups1270011288395367621r_real @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups1270011288395367621r_real @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8729_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8730_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8731_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups6602215022474089585er_rat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups6602215022474089585er_rat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8732_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8733_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8734_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ A4 )
     => ( ( groups7237345082560585321er_nat @ G
          @ ( minus_2355218937544613996nteger @ A4
            @ ( collect_Code_integer
              @ ^ [X: code_integer] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups7237345082560585321er_nat @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8735_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5690904116761175830ex_int @ G
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_int ) ) ) )
        = ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_8736_sum__power__add,axiom,
    ! [X2: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8737_sum__power__add,axiom,
    ! [X2: code_integer,M: nat,I5: set_nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I4: nat] : ( power_8256067586552552935nteger @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X2 @ M ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8738_sum__power__add,axiom,
    ! [X2: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8739_sum__power__add,axiom,
    ! [X2: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8740_sum__power__add,axiom,
    ! [X2: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8741_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8742_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_8743_suminf__finite,axiom,
    ! [N3: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).

% suminf_finite
thf(fact_8744_suminf__finite,axiom,
    ! [N3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).

% suminf_finite
thf(fact_8745_suminf__finite,axiom,
    ! [N3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N3 )
     => ( ! [N4: nat] :
            ( ~ ( member_nat @ N4 @ N3 )
           => ( ( F @ N4 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).

% suminf_finite
thf(fact_8746_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8747_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8748_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8749_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8750_sum__pos2,axiom,
    ! [I5: set_Code_integer,I: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( member_Code_integer @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I2: code_integer] :
                ( ( member_Code_integer @ I2 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8751_sum__pos2,axiom,
    ! [I5: set_VEBT_VEBT,I: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( member_VEBT_VEBT @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8752_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: real] :
                ( ( member_real @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8753_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8754_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: int] :
                ( ( member_int @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8755_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_8756_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups2240296850493347238T_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8757_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8758_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups1270011288395367621r_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8759_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8760_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8761_sum__pos,axiom,
    ! [I5: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ I5 )
     => ( ( I5 != bot_bo8194388402131092736T_VEBT )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups136491112297645522BT_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8762_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8763_sum__pos,axiom,
    ! [I5: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ I5 )
     => ( ( I5 != bot_bo3990330152332043303nteger )
       => ( ! [I2: code_integer] :
              ( ( member_Code_integer @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups6602215022474089585er_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8764_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8765_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_8766_suminf__of__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( ( real_V1803761363581548252l_real @ ( suminf_real @ X9 ) )
        = ( suminf_real
          @ ^ [N: nat] : ( real_V1803761363581548252l_real @ ( X9 @ N ) ) ) ) ) ).

% suminf_of_real
thf(fact_8767_suminf__of__real,axiom,
    ! [X9: nat > real] :
      ( ( summable_real @ X9 )
     => ( ( real_V4546457046886955230omplex @ ( suminf_real @ X9 ) )
        = ( suminf_complex
          @ ^ [N: nat] : ( real_V4546457046886955230omplex @ ( X9 @ N ) ) ) ) ) ).

% suminf_of_real
thf(fact_8768_norm__less__p1,axiom,
    ! [X2: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X2 ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8769_norm__less__p1,axiom,
    ! [X2: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X2 ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8770_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_VEBT_VEBT,S4: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T7 )
     => ( ( ord_le4337996190870823476T_VEBT @ S4 @ T7 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ T7 )
              = ( groups2240296850493347238T_real @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8771_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_real,S4: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T7 )
     => ( ( ord_less_eq_set_real @ S4 @ T7 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T7 )
              = ( groups8097168146408367636l_real @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8772_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_int,S4: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T7 )
              = ( groups8778361861064173332t_real @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8773_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T7 )
              = ( groups5808333547571424918x_real @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8774_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ T7 )
              = ( groups1270011288395367621r_real @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8775_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_VEBT_VEBT,S4: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T7 )
     => ( ( ord_le4337996190870823476T_VEBT @ S4 @ T7 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ T7 )
              = ( groups136491112297645522BT_rat @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8776_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_real,S4: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T7 )
     => ( ( ord_less_eq_set_real @ S4 @ T7 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T7 )
              = ( groups1300246762558778688al_rat @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8777_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_int,S4: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T7 )
              = ( groups3906332499630173760nt_rat @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8778_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T7 )
              = ( groups5058264527183730370ex_rat @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8779_sum_Omono__neutral__cong__right,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > rat,H2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups6602215022474089585er_rat @ G @ T7 )
              = ( groups6602215022474089585er_rat @ H2 @ S4 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_8780_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_VEBT_VEBT,S4: set_VEBT_VEBT,H2: vEBT_VEBT > real,G: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ T7 )
     => ( ( ord_le4337996190870823476T_VEBT @ S4 @ T7 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups2240296850493347238T_real @ G @ S4 )
              = ( groups2240296850493347238T_real @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8781_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_real,S4: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T7 )
     => ( ( ord_less_eq_set_real @ S4 @ T7 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S4 )
              = ( groups8097168146408367636l_real @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8782_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_int,S4: set_int,H2: int > real,G: int > real] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S4 )
              = ( groups8778361861064173332t_real @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8783_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_complex,S4: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S4 )
              = ( groups5808333547571424918x_real @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8784_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,H2: code_integer > real,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1270011288395367621r_real @ G @ S4 )
              = ( groups1270011288395367621r_real @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8785_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_VEBT_VEBT,S4: set_VEBT_VEBT,H2: vEBT_VEBT > rat,G: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ T7 )
     => ( ( ord_le4337996190870823476T_VEBT @ S4 @ T7 )
       => ( ! [X3: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups136491112297645522BT_rat @ G @ S4 )
              = ( groups136491112297645522BT_rat @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8786_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_real,S4: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T7 )
     => ( ( ord_less_eq_set_real @ S4 @ T7 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S4 )
              = ( groups1300246762558778688al_rat @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8787_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_int,S4: set_int,H2: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S4 )
              = ( groups3906332499630173760nt_rat @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8788_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_complex,S4: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S4 )
              = ( groups5058264527183730370ex_rat @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8789_sum_Omono__neutral__cong__left,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,H2: code_integer > rat,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( H2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: code_integer] :
                ( ( member_Code_integer @ X3 @ S4 )
               => ( ( G @ X3 )
                  = ( H2 @ X3 ) ) )
           => ( ( groups6602215022474089585er_rat @ G @ S4 )
              = ( groups6602215022474089585er_rat @ H2 @ T7 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_8790_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S4: set_int,G: int > real] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T7 )
            = ( groups8778361861064173332t_real @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8791_sum_Omono__neutral__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T7 )
            = ( groups5808333547571424918x_real @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8792_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ T7 )
            = ( groups1270011288395367621r_real @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8793_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S4: set_int,G: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T7 )
            = ( groups3906332499630173760nt_rat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8794_sum_Omono__neutral__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T7 )
            = ( groups5058264527183730370ex_rat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8795_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ T7 )
            = ( groups6602215022474089585er_rat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8796_sum_Omono__neutral__right,axiom,
    ! [T7: set_int,S4: set_int,G: int > nat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ T7 )
            = ( groups4541462559716669496nt_nat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8797_sum_Omono__neutral__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T7 )
            = ( groups5693394587270226106ex_nat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8798_sum_Omono__neutral__right,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ T7 )
            = ( groups7237345082560585321er_nat @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8799_sum_Omono__neutral__right,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T7 )
            = ( groups5690904116761175830ex_int @ G @ S4 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8800_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S4: set_int,G: int > real] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S4 )
            = ( groups8778361861064173332t_real @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8801_sum_Omono__neutral__left,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S4 )
            = ( groups5808333547571424918x_real @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8802_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > real] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_real ) )
         => ( ( groups1270011288395367621r_real @ G @ S4 )
            = ( groups1270011288395367621r_real @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8803_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S4: set_int,G: int > rat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S4 )
            = ( groups3906332499630173760nt_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8804_sum_Omono__neutral__left,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S4 )
            = ( groups5058264527183730370ex_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8805_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_rat ) )
         => ( ( groups6602215022474089585er_rat @ G @ S4 )
            = ( groups6602215022474089585er_rat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8806_sum_Omono__neutral__left,axiom,
    ! [T7: set_int,S4: set_int,G: int > nat] :
      ( ( finite_finite_int @ T7 )
     => ( ( ord_less_eq_set_int @ S4 @ T7 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G @ S4 )
            = ( groups4541462559716669496nt_nat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8807_sum_Omono__neutral__left,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S4 )
            = ( groups5693394587270226106ex_nat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8808_sum_Omono__neutral__left,axiom,
    ! [T7: set_Code_integer,S4: set_Code_integer,G: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ T7 )
     => ( ( ord_le7084787975880047091nteger @ S4 @ T7 )
       => ( ! [X3: code_integer] :
              ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_nat ) )
         => ( ( groups7237345082560585321er_nat @ G @ S4 )
            = ( groups7237345082560585321er_nat @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8809_sum_Omono__neutral__left,axiom,
    ! [T7: set_complex,S4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T7 )
     => ( ( ord_le211207098394363844omplex @ S4 @ T7 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T7 @ S4 ) )
             => ( ( G @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S4 )
            = ( groups5690904116761175830ex_int @ G @ T7 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8810_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B6: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B6 @ C4 )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) )
               => ( ( groups2240296850493347238T_real @ G @ A4 )
                  = ( groups2240296850493347238T_real @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8811_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B6: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B6 @ C4 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8812_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B6: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B6 @ C4 )
         => ( ! [A: int] :
                ( ( member_int @ A @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8813_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B6: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B6 @ C4 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8814_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B6: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B6 @ C4 )
         => ( ! [A: code_integer] :
                ( ( member_Code_integer @ A @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) )
               => ( ( groups1270011288395367621r_real @ G @ A4 )
                  = ( groups1270011288395367621r_real @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8815_sum_Osame__carrierI,axiom,
    ! [C4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B6: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B6 @ C4 )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) )
               => ( ( groups136491112297645522BT_rat @ G @ A4 )
                  = ( groups136491112297645522BT_rat @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8816_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B6: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B6 @ C4 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8817_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B6: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B6 @ C4 )
         => ( ! [A: int] :
                ( ( member_int @ A @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8818_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B6: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B6 @ C4 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8819_sum_Osame__carrierI,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B6: set_Code_integer,G: code_integer > rat,H2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B6 @ C4 )
         => ( ! [A: code_integer] :
                ( ( member_Code_integer @ A @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ C4 )
                  = ( groups6602215022474089585er_rat @ H2 @ C4 ) )
               => ( ( groups6602215022474089585er_rat @ G @ A4 )
                  = ( groups6602215022474089585er_rat @ H2 @ B6 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8820_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B6: set_VEBT_VEBT,G: vEBT_VEBT > real,H2: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B6 @ C4 )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups2240296850493347238T_real @ G @ A4 )
                  = ( groups2240296850493347238T_real @ H2 @ B6 ) )
                = ( ( groups2240296850493347238T_real @ G @ C4 )
                  = ( groups2240296850493347238T_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8821_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B6: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B6 @ C4 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H2 @ B6 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8822_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B6: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B6 @ C4 )
         => ( ! [A: int] :
                ( ( member_int @ A @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H2 @ B6 ) )
                = ( ( groups8778361861064173332t_real @ G @ C4 )
                  = ( groups8778361861064173332t_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8823_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B6: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B6 @ C4 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H2 @ B6 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8824_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B6: set_Code_integer,G: code_integer > real,H2: code_integer > real] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B6 @ C4 )
         => ( ! [A: code_integer] :
                ( ( member_Code_integer @ A @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_real ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_real ) )
             => ( ( ( groups1270011288395367621r_real @ G @ A4 )
                  = ( groups1270011288395367621r_real @ H2 @ B6 ) )
                = ( ( groups1270011288395367621r_real @ G @ C4 )
                  = ( groups1270011288395367621r_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8825_sum_Osame__carrier,axiom,
    ! [C4: set_VEBT_VEBT,A4: set_VEBT_VEBT,B6: set_VEBT_VEBT,G: vEBT_VEBT > rat,H2: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ C4 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ C4 )
       => ( ( ord_le4337996190870823476T_VEBT @ B6 @ C4 )
         => ( ! [A: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ A @ ( minus_5127226145743854075T_VEBT @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups136491112297645522BT_rat @ G @ A4 )
                  = ( groups136491112297645522BT_rat @ H2 @ B6 ) )
                = ( ( groups136491112297645522BT_rat @ G @ C4 )
                  = ( groups136491112297645522BT_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8826_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B6: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B6 @ C4 )
         => ( ! [A: real] :
                ( ( member_real @ A @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: real] :
                  ( ( member_real @ B @ ( minus_minus_set_real @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H2 @ B6 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8827_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B6: set_int,G: int > rat,H2: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B6 @ C4 )
         => ( ! [A: int] :
                ( ( member_int @ A @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: int] :
                  ( ( member_int @ B @ ( minus_minus_set_int @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ B6 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C4 )
                  = ( groups3906332499630173760nt_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8828_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B6: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B6 @ C4 )
         => ( ! [A: complex] :
                ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: complex] :
                  ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B6 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8829_sum_Osame__carrier,axiom,
    ! [C4: set_Code_integer,A4: set_Code_integer,B6: set_Code_integer,G: code_integer > rat,H2: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ C4 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ C4 )
       => ( ( ord_le7084787975880047091nteger @ B6 @ C4 )
         => ( ! [A: code_integer] :
                ( ( member_Code_integer @ A @ ( minus_2355218937544613996nteger @ C4 @ A4 ) )
               => ( ( G @ A )
                  = zero_zero_rat ) )
           => ( ! [B: code_integer] :
                  ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ C4 @ B6 ) )
                 => ( ( H2 @ B )
                    = zero_zero_rat ) )
             => ( ( ( groups6602215022474089585er_rat @ G @ A4 )
                  = ( groups6602215022474089585er_rat @ H2 @ B6 ) )
                = ( ( groups6602215022474089585er_rat @ G @ C4 )
                  = ( groups6602215022474089585er_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8830_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > word_N3645301735248828278l_num1,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_z3563351764282998399l_num1 )
     => ( ( groups2996710295995929986l_num1 @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2996710295995929986l_num1 @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8831_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_8832_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_8833_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_8834_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_8835_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8836_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8837_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8838_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_8839_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8840_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8841_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8842_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8843_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8844_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8845_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8846_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8847_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8848_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8849_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8850_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8851_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8852_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8853_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_8854_sum__mono2,axiom,
    ! [B6: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B6 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B6 )
       => ( ! [B: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B6 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8855_sum__mono2,axiom,
    ! [B6: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B6 )
     => ( ( ord_less_eq_set_real @ A4 @ B6 )
       => ( ! [B: real] :
              ( ( member_real @ B @ ( minus_minus_set_real @ B6 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8856_sum__mono2,axiom,
    ! [B6: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B6 )
     => ( ( ord_less_eq_set_int @ A4 @ B6 )
       => ( ! [B: int] :
              ( ( member_int @ B @ ( minus_minus_set_int @ B6 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8857_sum__mono2,axiom,
    ! [B6: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B6 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B6 )
       => ( ! [B: complex] :
              ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B6 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8858_sum__mono2,axiom,
    ! [B6: set_Code_integer,A4: set_Code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B6 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B6 )
       => ( ! [B: code_integer] :
              ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B6 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
         => ( ord_less_eq_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8859_sum__mono2,axiom,
    ! [B6: set_VEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B6 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B6 )
       => ( ! [B: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ B @ ( minus_5127226145743854075T_VEBT @ B6 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8860_sum__mono2,axiom,
    ! [B6: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B6 )
     => ( ( ord_less_eq_set_real @ A4 @ B6 )
       => ( ! [B: real] :
              ( ( member_real @ B @ ( minus_minus_set_real @ B6 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8861_sum__mono2,axiom,
    ! [B6: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B6 )
     => ( ( ord_less_eq_set_int @ A4 @ B6 )
       => ( ! [B: int] :
              ( ( member_int @ B @ ( minus_minus_set_int @ B6 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8862_sum__mono2,axiom,
    ! [B6: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B6 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B6 )
       => ( ! [B: complex] :
              ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B6 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8863_sum__mono2,axiom,
    ! [B6: set_Code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B6 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B6 )
       => ( ! [B: code_integer] :
              ( ( member_Code_integer @ B @ ( minus_2355218937544613996nteger @ B6 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
         => ( ord_less_eq_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B6 ) ) ) ) ) ).

% sum_mono2
thf(fact_8864_arccos__lbound,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) ) ) ) ).

% arccos_lbound
thf(fact_8865_arccos__less__arccos,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ( ord_less_eq_real @ Y2 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y2 ) @ ( arccos @ X2 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_8866_arccos__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y2 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y2 ) )
          = ( ord_less_real @ Y2 @ X2 ) ) ) ) ).

% arccos_less_mono
thf(fact_8867_arccos__cos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ pi )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = X2 ) ) ) ).

% arccos_cos
thf(fact_8868_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8869_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8870_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8871_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8872_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8873_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8874_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N4: nat] :
              ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8875_sum_Odelta__remove,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > real,C: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_real @ ( B3 @ A3 ) @ ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups2240296850493347238T_real
              @ ^ [K3: vEBT_VEBT] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups2240296850493347238T_real @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8876_sum_Odelta__remove,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_real @ ( B3 @ A3 ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8877_sum_Odelta__remove,axiom,
    ! [S4: set_Code_integer,A3: code_integer,B3: code_integer > real,C: code_integer > real] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_real @ ( B3 @ A3 ) @ ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A3 @ S4 )
         => ( ( groups1270011288395367621r_real
              @ ^ [K3: code_integer] : ( if_real @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups1270011288395367621r_real @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8878_sum_Odelta__remove,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > rat,C: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_rat @ ( B3 @ A3 ) @ ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups136491112297645522BT_rat
              @ ^ [K3: vEBT_VEBT] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups136491112297645522BT_rat @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8879_sum_Odelta__remove,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_rat @ ( B3 @ A3 ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8880_sum_Odelta__remove,axiom,
    ! [S4: set_Code_integer,A3: code_integer,B3: code_integer > rat,C: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( ( member_Code_integer @ A3 @ S4 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_rat @ ( B3 @ A3 ) @ ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A3 @ S4 )
         => ( ( groups6602215022474089585er_rat
              @ ^ [K3: code_integer] : ( if_rat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups6602215022474089585er_rat @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8881_sum_Odelta__remove,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > nat,C: vEBT_VEBT > nat] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_nat @ ( B3 @ A3 ) @ ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups771621172384141258BT_nat
              @ ^ [K3: vEBT_VEBT] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups771621172384141258BT_nat @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8882_sum_Odelta__remove,axiom,
    ! [S4: set_complex,A3: complex,B3: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S4 )
     => ( ( ( member_complex @ A3 @ S4 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_nat @ ( B3 @ A3 ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A3 @ S4 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S4 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8883_sum_Odelta__remove,axiom,
    ! [S4: set_Code_integer,A3: code_integer,B3: code_integer > nat,C: code_integer > nat] :
      ( ( finite6017078050557962740nteger @ S4 )
     => ( ( ( member_Code_integer @ A3 @ S4 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_nat @ ( B3 @ A3 ) @ ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) )
        & ( ~ ( member_Code_integer @ A3 @ S4 )
         => ( ( groups7237345082560585321er_nat
              @ ^ [K3: code_integer] : ( if_nat @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups7237345082560585321er_nat @ C @ ( minus_2355218937544613996nteger @ S4 @ ( insert_Code_integer @ A3 @ bot_bo3990330152332043303nteger ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8884_sum_Odelta__remove,axiom,
    ! [S4: set_VEBT_VEBT,A3: vEBT_VEBT,B3: vEBT_VEBT > int,C: vEBT_VEBT > int] :
      ( ( finite5795047828879050333T_VEBT @ S4 )
     => ( ( ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K3: vEBT_VEBT] : ( if_int @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( plus_plus_int @ ( B3 @ A3 ) @ ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) )
        & ( ~ ( member_VEBT_VEBT @ A3 @ S4 )
         => ( ( groups769130701875090982BT_int
              @ ^ [K3: vEBT_VEBT] : ( if_int @ ( K3 = A3 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
              @ S4 )
            = ( groups769130701875090982BT_int @ C @ ( minus_5127226145743854075T_VEBT @ S4 @ ( insert_VEBT_VEBT @ A3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8885_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8886_sum__strict__mono2,axiom,
    ! [B6: set_VEBT_VEBT,A4: set_VEBT_VEBT,B3: vEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( finite5795047828879050333T_VEBT @ B6 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B6 )
       => ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B6 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B6 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups2240296850493347238T_real @ F @ A4 ) @ ( groups2240296850493347238T_real @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8887_sum__strict__mono2,axiom,
    ! [B6: set_real,A4: set_real,B3: real,F: real > real] :
      ( ( finite_finite_real @ B6 )
     => ( ( ord_less_eq_set_real @ A4 @ B6 )
       => ( ( member_real @ B3 @ ( minus_minus_set_real @ B6 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B6 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8888_sum__strict__mono2,axiom,
    ! [B6: set_int,A4: set_int,B3: int,F: int > real] :
      ( ( finite_finite_int @ B6 )
     => ( ( ord_less_eq_set_int @ A4 @ B6 )
       => ( ( member_int @ B3 @ ( minus_minus_set_int @ B6 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B6 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8889_sum__strict__mono2,axiom,
    ! [B6: set_complex,A4: set_complex,B3: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B6 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B6 )
       => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B6 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B6 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8890_sum__strict__mono2,axiom,
    ! [B6: set_Code_integer,A4: set_Code_integer,B3: code_integer,F: code_integer > real] :
      ( ( finite6017078050557962740nteger @ B6 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B6 )
       => ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B6 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B6 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
             => ( ord_less_real @ ( groups1270011288395367621r_real @ F @ A4 ) @ ( groups1270011288395367621r_real @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8891_sum__strict__mono2,axiom,
    ! [B6: set_VEBT_VEBT,A4: set_VEBT_VEBT,B3: vEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( finite5795047828879050333T_VEBT @ B6 )
     => ( ( ord_le4337996190870823476T_VEBT @ A4 @ B6 )
       => ( ( member_VEBT_VEBT @ B3 @ ( minus_5127226145743854075T_VEBT @ B6 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X3: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X3 @ B6 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups136491112297645522BT_rat @ F @ A4 ) @ ( groups136491112297645522BT_rat @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8892_sum__strict__mono2,axiom,
    ! [B6: set_real,A4: set_real,B3: real,F: real > rat] :
      ( ( finite_finite_real @ B6 )
     => ( ( ord_less_eq_set_real @ A4 @ B6 )
       => ( ( member_real @ B3 @ ( minus_minus_set_real @ B6 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B6 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8893_sum__strict__mono2,axiom,
    ! [B6: set_int,A4: set_int,B3: int,F: int > rat] :
      ( ( finite_finite_int @ B6 )
     => ( ( ord_less_eq_set_int @ A4 @ B6 )
       => ( ( member_int @ B3 @ ( minus_minus_set_int @ B6 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B6 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8894_sum__strict__mono2,axiom,
    ! [B6: set_complex,A4: set_complex,B3: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B6 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B6 )
       => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B6 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B6 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8895_sum__strict__mono2,axiom,
    ! [B6: set_Code_integer,A4: set_Code_integer,B3: code_integer,F: code_integer > rat] :
      ( ( finite6017078050557962740nteger @ B6 )
     => ( ( ord_le7084787975880047091nteger @ A4 @ B6 )
       => ( ( member_Code_integer @ B3 @ ( minus_2355218937544613996nteger @ B6 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
           => ( ! [X3: code_integer] :
                  ( ( member_Code_integer @ X3 @ B6 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
             => ( ord_less_rat @ ( groups6602215022474089585er_rat @ F @ A4 ) @ ( groups6602215022474089585er_rat @ F @ B6 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8896_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > real] :
      ( ( member_VEBT_VEBT @ I @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups2240296850493347238T_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8897_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8898_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: code_integer > real] :
      ( ( member_Code_integer @ I @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups1270011288395367621r_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8899_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > real] :
      ( ( member_real @ I @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8900_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: int > real] :
      ( ( member_int @ I @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8901_member__le__sum,axiom,
    ! [I: vEBT_VEBT,A4: set_VEBT_VEBT,F: vEBT_VEBT > rat] :
      ( ( member_VEBT_VEBT @ I @ A4 )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( minus_5127226145743854075T_VEBT @ A4 @ ( insert_VEBT_VEBT @ I @ bot_bo8194388402131092736T_VEBT ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite5795047828879050333T_VEBT @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups136491112297645522BT_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8902_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8903_member__le__sum,axiom,
    ! [I: code_integer,A4: set_Code_integer,F: code_integer > rat] :
      ( ( member_Code_integer @ I @ A4 )
     => ( ! [X3: code_integer] :
            ( ( member_Code_integer @ X3 @ ( minus_2355218937544613996nteger @ A4 @ ( insert_Code_integer @ I @ bot_bo3990330152332043303nteger ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite6017078050557962740nteger @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups6602215022474089585er_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8904_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > rat] :
      ( ( member_real @ I @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8905_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: int > rat] :
      ( ( member_int @ I @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8906_convex__sum__bound__le,axiom,
    ! [I5: set_VEBT_VEBT,X2: vEBT_VEBT > int,A3: vEBT_VEBT > int,B3: int,Delta: int] :
      ( ! [I2: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ I2 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X2 @ I2 ) ) )
     => ( ( ( groups769130701875090982BT_int @ X2 @ I5 )
          = one_one_int )
       => ( ! [I2: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ I2 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups769130701875090982BT_int
                  @ ^ [I4: vEBT_VEBT] : ( times_times_int @ ( A3 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8907_convex__sum__bound__le,axiom,
    ! [I5: set_real,X2: real > int,A3: real > int,B3: int,Delta: int] :
      ( ! [I2: real] :
          ( ( member_real @ I2 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X2 @ I2 ) ) )
     => ( ( ( groups1932886352136224148al_int @ X2 @ I5 )
          = one_one_int )
       => ( ! [I2: real] :
              ( ( member_real @ I2 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups1932886352136224148al_int
                  @ ^ [I4: real] : ( times_times_int @ ( A3 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8908_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X2: nat > real,A3: nat > real,B3: real,Delta: real] :
      ( ! [I2: nat] :
          ( ( member_nat @ I2 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
     => ( ( ( groups6591440286371151544t_real @ X2 @ I5 )
          = one_one_real )
       => ( ! [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( A3 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8909_convex__sum__bound__le,axiom,
    ! [I5: set_int,X2: int > int,A3: int > int,B3: int,Delta: int] :
      ( ! [I2: int] :
          ( ( member_int @ I2 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X2 @ I2 ) ) )
     => ( ( ( groups4538972089207619220nt_int @ X2 @ I5 )
          = one_one_int )
       => ( ! [I2: int] :
              ( ( member_int @ I2 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A3 @ I2 ) @ B3 ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups4538972089207619220nt_int
                  @ ^ [I4: int] : ( times_times_int @ ( A3 @ I4 ) @ ( X2 @ I4 ) )
                  @ I5 )
                @ B3 ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8910_arccos__lt__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_real @ Y2 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_8911_arccos__bounded,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_8912_sin__arccos__nonzero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_8913_arccos__cos2,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( arccos @ ( cos_real @ X2 ) )
          = ( uminus_uminus_real @ X2 ) ) ) ) ).

% arccos_cos2
thf(fact_8914_arccos,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y2 ) )
          & ( ord_less_eq_real @ ( arccos @ Y2 ) @ pi )
          & ( ( cos_real @ ( arccos @ Y2 ) )
            = Y2 ) ) ) ) ).

% arccos
thf(fact_8915_mask__eq__sum__exp__nat,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
      = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( collect_nat
          @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_8916_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_nat
thf(fact_8917_arith__series__nat,axiom,
    ! [A3: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ A3 @ ( times_times_nat @ I4 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_8918_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8919_arccos__le__pi2,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( ord_less_eq_real @ Y2 @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_8920_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_8921_sin__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X2 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_8922_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8923_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8924_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_8925_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8926_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8927_power__half__series,axiom,
    ( sums_real
    @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8928_sums__if_H,axiom,
    ! [G: nat > real,X2: real] :
      ( ( sums_real @ G @ X2 )
     => ( sums_real
        @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X2 ) ) ).

% sums_if'
thf(fact_8929_sums__if,axiom,
    ! [G: nat > real,X2: real,F: nat > real,Y2: real] :
      ( ( sums_real @ G @ X2 )
     => ( ( sums_real @ F @ Y2 )
       => ( sums_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( F @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X2 @ Y2 ) ) ) ) ).

% sums_if
thf(fact_8930_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_8931_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( set_or1266510415728281911st_int @ M @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8932_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D2: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D2 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8933_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8934_sin__paired,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X2 ) ) ).

% sin_paired
thf(fact_8935_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8936_square__fact__le__2__fact,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% square_fact_le_2_fact
thf(fact_8937_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8938_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B9: 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 @ N2 ) )
            @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8939_Maclaurin__cos__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
      & ( ( cos_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8940_Maclaurin__cos__expansion2,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X2 )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8941_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ? [T3: real] :
            ( ( ord_less_real @ X2 @ T3 )
            & ( ord_less_real @ T3 @ zero_zero_real )
            & ( ( cos_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8942_cos__paired,axiom,
    ! [X2: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_real @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      @ ( cos_real @ X2 ) ) ).

% cos_paired
thf(fact_8943_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X2 )
            & ( ( sin_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8944_Maclaurin__sin__expansion4,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ? [T3: real] :
          ( ( ord_less_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ X2 )
          & ( ( sin_real @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8945_Maclaurin__sin__expansion2,axiom,
    ! [X2: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
      & ( ( sin_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8946_Maclaurin__sin__expansion,axiom,
    ! [X2: real,N2: nat] :
    ? [T3: real] :
      ( ( sin_real @ X2 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8947_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8948_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_8949_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_8950_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8951_fact__ge__Suc__0__nat,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_Suc_0_nat
thf(fact_8952_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_8953_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8954_fact__div__fact__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R ) ) ) @ ( power_power_nat @ N2 @ R ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8955_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8956_Maclaurin__exp__lt,axiom,
    ! [X2: real,N2: nat] :
      ( ( X2 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
            & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
            & ( ( exp_real @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8957_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      ( ( vEBT_VEBT_height @ ( vEBT_Leaf @ A3 @ B3 ) )
      = zero_zero_nat ) ).

% VEBT_internal.height.simps(1)
thf(fact_8958_exp__less__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ X2 @ Y2 )
     => ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) ) ) ).

% exp_less_mono
thf(fact_8959_exp__less__cancel__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% exp_less_cancel_iff
thf(fact_8960_exp__eq__one__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ X2 )
        = one_one_real )
      = ( X2 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_8961_exp__less__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_8962_one__less__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% one_less_exp_iff
thf(fact_8963_exp__le__one__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_8964_one__le__exp__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% one_le_exp_iff
thf(fact_8965_exp__ln__iff,axiom,
    ! [X2: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% exp_ln_iff
thf(fact_8966_exp__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( exp_real @ ( ln_ln_real @ X2 ) )
        = X2 ) ) ).

% exp_ln
thf(fact_8967_exp__less__cancel,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y2 ) )
     => ( ord_less_real @ X2 @ Y2 ) ) ).

% exp_less_cancel
thf(fact_8968_exp__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y2 )
     => ? [X3: real] :
          ( ( exp_real @ X3 )
          = Y2 ) ) ).

% exp_total
thf(fact_8969_exp__gt__zero,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_gt_zero
thf(fact_8970_not__exp__less__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_8971_not__exp__le__zero,axiom,
    ! [X2: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_8972_exp__ge__zero,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).

% exp_ge_zero
thf(fact_8973_exp__gt__one,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).

% exp_gt_one
thf(fact_8974_exp__ge__add__one__self__aux,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_8975_lemma__exp__total,axiom,
    ! [Y2: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y2 )
     => ? [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
          & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y2 @ one_one_real ) )
          & ( ( exp_real @ X3 )
            = Y2 ) ) ) ).

% lemma_exp_total
thf(fact_8976_ln__ge__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ Y2 @ ( ln_ln_real @ X2 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y2 ) @ X2 ) ) ) ).

% ln_ge_iff
thf(fact_8977_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_8978_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_8979_VEBT__internal_Oheight_Ocases,axiom,
    ! [X2: vEBT_VEBT] :
      ( ! [A: $o,B: $o] :
          ( X2
         != ( vEBT_Leaf @ A @ B ) )
     => ~ ! [Uu3: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
            ( X2
           != ( vEBT_Node @ Uu3 @ Deg @ TreeList3 @ Summary ) ) ) ).

% VEBT_internal.height.cases
thf(fact_8980_exp__bound,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_8981_real__exp__bound__lemma,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_8982_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X2 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8983_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8984_Maclaurin__exp__le,axiom,
    ! [X2: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
      & ( ( exp_real @ X2 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8985_exp__lower__Taylor__quadratic,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_8986_log__base__10__eq2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% log_base_10_eq2
thf(fact_8987_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_8988_log__base__10__eq1,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).

% log_base_10_eq1
thf(fact_8989_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8990_floor__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A3 ) @ ( numeral_numeral_real @ B3 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A3 ) @ ( numeral_numeral_int @ B3 ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8991_floor__one__divide__eq__div__numeral,axiom,
    ! [B3: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B3 ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B3 ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8992_floor__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A3 ) @ ( numeral_numeral_real @ B3 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A3 ) ) @ ( numeral_numeral_int @ B3 ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8993_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B3: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B3 ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B3 ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8994_floor__eq,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N2 ) ) ) ).

% floor_eq
thf(fact_8995_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_8996_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_8997_floor__eq2,axiom,
    ! [N2: int,X2: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X2 )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_8998_floor__divide__real__eq__div,axiom,
    ! [B3: int,A3: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B3 )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A3 @ ( ring_1_of_int_real @ B3 ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A3 ) @ B3 ) ) ) ).

% floor_divide_real_eq_div
thf(fact_8999_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9000_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9001_floor__log__eq__powr__iff,axiom,
    ! [X2: real,B3: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ one_one_real @ B3 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B3 @ X2 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B3 @ ( ring_1_of_int_real @ K ) ) @ X2 )
            & ( ord_less_real @ X2 @ ( powr_real @ B3 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_9002_floor__log2__div2,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_9003_floor__log__nat__eq__if,axiom,
    ! [B3: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_9004_Maclaurin__sin__bound,axiom,
    ! [X2: real,N2: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X2 )
          @ ( groups6591440286371151544t_real
            @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X2 @ M3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N2 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9005__092_060open_062heap_Oadmissible_A_I_092_060lambda_062vebt__predi_H_O_A_092_060forall_062x_Axa_Axb_O_Arefines_A_Ivebt__predi_Axa_Axb_J_A_Icurry_A_Icurry_Avebt__predi_H_J_Ax_Axa_Axb_J_J_092_060close_062,axiom,
    ( comple1540308706681863803on_nat @ ( partia6039416512482706817Ti_nat @ heap_T7048022066654196708on_nat ) @ ( partia7778075537949639097Ti_nat @ heap_T7875578835903804844on_nat )
    @ ^ [Vebt_predi3: produc3960855945107176009Ti_nat > heap_T2636463487746394924on_nat] :
      ! [X: vEBT_VEBT,Y: vEBT_VEBTi,Z3: nat] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Y @ Z3 ) @ ( produc1114182431767986483on_nat @ ( produc1757988346207259447on_nat @ Vebt_predi3 ) @ X @ Y @ Z3 ) ) ) ).

% \<open>heap.admissible (\<lambda>vebt_predi'. \<forall>x xa xb. refines (vebt_predi xa xb) (curry (curry vebt_predi') x xa xb))\<close>
thf(fact_9006_inverse__powr,axiom,
    ! [Y2: real,A3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
     => ( ( powr_real @ ( inverse_inverse_real @ Y2 ) @ A3 )
        = ( inverse_inverse_real @ ( powr_real @ Y2 @ A3 ) ) ) ) ).

% inverse_powr
thf(fact_9007_prod__int__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9008_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P @ D2 )
           => ( P @ E ) ) )
     => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9009_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D2: real,E: real] :
          ( ( ord_less_real @ D2 @ E )
         => ( ( P @ D2 )
           => ( P @ E ) ) )
     => ( ! [N4: nat] :
            ( ( N4 != zero_zero_nat )
           => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( P @ E2 ) ) ) ) ).

% forall_pos_mono
thf(fact_9010_real__arch__inverse,axiom,
    ! [E2: real] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
      = ( ? [N: nat] :
            ( ( N != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).

% real_arch_inverse
thf(fact_9011_sqrt__divide__self__eq,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
        = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_9012_ln__inverse,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X2 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X2 ) ) ) ) ).

% ln_inverse
thf(fact_9013_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9014_log__inverse,axiom,
    ! [A3: real,X2: real] :
      ( ( ord_less_real @ zero_zero_real @ A3 )
     => ( ( A3 != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( log @ A3 @ ( inverse_inverse_real @ X2 ) )
            = ( uminus_uminus_real @ ( log @ A3 @ X2 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9015_exp__plus__inverse__exp,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_9016_plus__inverse__ge__2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9017_real__inv__sqrt__pow2,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X2 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9018_tan__cot,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
      = ( inverse_inverse_real @ ( tan_real @ X2 ) ) ) ).

% tan_cot
thf(fact_9019_real__le__x__sinh,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_9020_real__le__abs__sinh,axiom,
    ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_9021_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_9022_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_9023_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_9024_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_9025_binomial__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( binomial @ N2 @ K )
        = zero_zero_nat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% binomial_eq_0_iff
thf(fact_9026_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_9027_zero__less__binomial__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
      = ( ord_less_eq_nat @ K @ N2 ) ) ).

% zero_less_binomial_iff
thf(fact_9028_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9029_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9030_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_9031_sum__choose__lower,axiom,
    ! [R: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_9032_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9033_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9034_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_9035_binomial__symmetric,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% binomial_symmetric
thf(fact_9036_binomial__le__pow,axiom,
    ! [R: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R ) @ ( power_power_nat @ N2 @ R ) ) ) ).

% binomial_le_pow
thf(fact_9037_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9038_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9039_vandermonde,axiom,
    ! [M: nat,N2: nat,R: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R ) ) ).

% vandermonde
thf(fact_9040_choose__row__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% choose_row_sum
thf(fact_9041_binomial,axiom,
    ! [A3: nat,B3: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A3 @ B3 ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A3 @ K3 ) ) @ ( power_power_nat @ B3 @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_9042_zero__less__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).

% zero_less_binomial
thf(fact_9043_binomial__Suc__Suc__eq__times,axiom,
    ! [N2: nat,K: nat] :
      ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_9044_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_9045_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_9046_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% choose_square_sum
thf(fact_9047_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_9048_binomial__absorption,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorption
thf(fact_9049_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_9050_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N2 @ I4 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_9051_binomial__fact__lemma,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
        = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% binomial_fact_lemma
thf(fact_9052_binomial__maximum,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_9053_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
       => ( ( ord_less_eq_nat @ K4 @ N2 )
         => ( ord_less_eq_nat @ ( binomial @ N2 @ K4 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_antimono
thf(fact_9054_binomial__mono,axiom,
    ! [K: nat,K4: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N2 )
       => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K4 ) ) ) ) ).

% binomial_mono
thf(fact_9055_binomial__maximum_H,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% binomial_maximum'
thf(fact_9056_binomial__le__pow2,axiom,
    ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% binomial_le_pow2
thf(fact_9057_choose__reduce__nat,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( binomial @ N2 @ K )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_9058_times__binomial__minus1__eq,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_9059_binomial__altdef__nat,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( binomial @ N2 @ K )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_9060_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_9061_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K4 ) @ N2 )
       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K4 ) ) ) ) ).

% binomial_strict_mono
thf(fact_9062_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N2: nat] :
      ( ( ord_less_nat @ K @ K4 )
     => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
       => ( ( ord_less_eq_nat @ K4 @ N2 )
         => ( ord_less_nat @ ( binomial @ N2 @ K4 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_9063_central__binomial__odd,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% central_binomial_odd
thf(fact_9064_binomial__addition__formula,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( binomial @ N2 @ ( suc @ K ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).

% binomial_addition_formula
thf(fact_9065_atLeast1__atMost__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9066_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_9067_polynomial__product__nat,axiom,
    ! [M: nat,A3: nat > nat,N2: nat,B3: nat > nat,X2: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ M @ I2 )
         => ( ( A3 @ I2 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B3 @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A3 @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B3 @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R6: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A3 @ K3 ) @ ( B3 @ ( minus_minus_nat @ R6 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R6 ) )
                @ ( power_power_nat @ X2 @ R6 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9068_central__binomial__lower__bound,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).

% central_binomial_lower_bound
thf(fact_9069_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9070_sinh__real__zero__iff,axiom,
    ! [X2: real] :
      ( ( ( sinh_real @ X2 )
        = zero_zero_real )
      = ( X2 = zero_zero_real ) ) ).

% sinh_real_zero_iff
thf(fact_9071_sinh__real__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y2 ) )
      = ( ord_less_real @ X2 @ Y2 ) ) ).

% sinh_real_less_iff
thf(fact_9072_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_9073_sinh__real__neg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9074_sinh__real__pos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_pos_iff
thf(fact_9075_sinh__real__nonneg__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% sinh_real_nonneg_iff
thf(fact_9076_sinh__real__nonpos__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_9077_sinh__less__cosh__real,axiom,
    ! [X2: real] : ( ord_less_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).

% sinh_less_cosh_real
thf(fact_9078_cosh__real__nonzero,axiom,
    ! [X2: real] :
      ( ( cosh_real @ X2 )
     != zero_zero_real ) ).

% cosh_real_nonzero
thf(fact_9079_Complex__eq__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ( complex2 @ A3 @ B3 )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A3
          = ( numeral_numeral_real @ W ) )
        & ( B3 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_9080_cosh__real__pos,axiom,
    ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_pos
thf(fact_9081_cosh__real__nonpos__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_9082_cosh__real__nonneg__le__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_9083_cosh__real__nonneg,axiom,
    ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).

% cosh_real_nonneg
thf(fact_9084_complex__eq__cancel__iff2,axiom,
    ! [X2: real,Y2: real,Xa: real] :
      ( ( ( complex2 @ X2 @ Y2 )
        = ( real_V4546457046886955230omplex @ Xa ) )
      = ( ( X2 = Xa )
        & ( Y2 = zero_zero_real ) ) ) ).

% complex_eq_cancel_iff2
thf(fact_9085_complex__of__real__code,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [X: real] : ( complex2 @ X @ zero_zero_real ) ) ) ).

% complex_of_real_code
thf(fact_9086_complex__of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R6: real] : ( complex2 @ R6 @ zero_zero_real ) ) ) ).

% complex_of_real_def
thf(fact_9087_zero__complex_Ocode,axiom,
    ( zero_zero_complex
    = ( complex2 @ zero_zero_real @ zero_zero_real ) ) ).

% zero_complex.code
thf(fact_9088_Complex__eq__0,axiom,
    ! [A3: real,B3: real] :
      ( ( ( complex2 @ A3 @ B3 )
        = zero_zero_complex )
      = ( ( A3 = zero_zero_real )
        & ( B3 = zero_zero_real ) ) ) ).

% Complex_eq_0
thf(fact_9089_cosh__real__strict__mono,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9090_cosh__real__nonneg__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_real @ X2 @ Y2 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9091_cosh__real__nonpos__less__iff,axiom,
    ! [X2: real,Y2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y2 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y2 ) )
          = ( ord_less_real @ Y2 @ X2 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9092_Complex__eq__neg__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( ( complex2 @ A3 @ B3 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A3
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B3 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_9093_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_9094_Complex__eq__1,axiom,
    ! [A3: real,B3: real] :
      ( ( ( complex2 @ A3 @ B3 )
        = one_one_complex )
      = ( ( A3 = one_one_real )
        & ( B3 = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_9095_arcosh__cosh__real,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( arcosh_real @ ( cosh_real @ X2 ) )
        = X2 ) ) ).

% arcosh_cosh_real
thf(fact_9096_Complex__eq__neg__1,axiom,
    ! [A3: real,B3: real] :
      ( ( ( complex2 @ A3 @ B3 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A3
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B3 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_9097_complex__norm,axiom,
    ! [X2: real,Y2: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X2 @ Y2 ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_9098_complex__inverse,axiom,
    ! [A3: real,B3: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A3 @ B3 ) )
      = ( complex2 @ ( divide_divide_real @ A3 @ ( plus_plus_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B3 ) @ ( plus_plus_real @ ( power_power_real @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9099_cosh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9100_sinh__ln__real,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9101_cot__less__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
     => ( ( ord_less_real @ X2 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9102_cot__pi,axiom,
    ( ( cot_real @ pi )
    = zero_zero_real ) ).

% cot_pi
thf(fact_9103_cot__npi,axiom,
    ! [N2: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_9104_cot__periodic,axiom,
    ! [X2: real] :
      ( ( cot_real @ ( plus_plus_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X2 ) ) ).

% cot_periodic
thf(fact_9105_cot__gt__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).

% cot_gt_zero
thf(fact_9106_tan__cot_H,axiom,
    ! [X2: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) )
      = ( cot_real @ X2 ) ) ).

% tan_cot'
thf(fact_9107_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_9108_i__even__power,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).

% i_even_power
thf(fact_9109_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_9110_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9111_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_9112_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_9113_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero_complex ).

% complex_i_not_zero
thf(fact_9114_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_9115_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_9116_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_9117_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_9118_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_9119_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_9120_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_9121_Complex__eq__i,axiom,
    ! [X2: real,Y2: real] :
      ( ( ( complex2 @ X2 @ Y2 )
        = imaginary_unit )
      = ( ( X2 = zero_zero_real )
        & ( Y2 = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_9122_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_9123_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_9124_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_9125_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_9126_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9127_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( ( csqrt @ Z )
        = zero_zero_complex )
      = ( Z = zero_zero_complex ) ) ).

% csqrt_eq_0
thf(fact_9128_csqrt__0,axiom,
    ( ( csqrt @ zero_zero_complex )
    = zero_zero_complex ) ).

% csqrt_0
thf(fact_9129_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_9130_Arg__zero,axiom,
    ( ( arg @ zero_zero_complex )
    = zero_zero_real ) ).

% Arg_zero
thf(fact_9131_of__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).

% of_real_sqrt
thf(fact_9132_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_9133_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_9134_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D3: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D3 @ Z6 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9135_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D3: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z6: int,Z3: int] :
                ( ( ord_less_eq_int @ D3 @ Z3 )
                & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9136_cis__zero,axiom,
    ( ( cis @ zero_zero_real )
    = one_one_complex ) ).

% cis_zero
thf(fact_9137_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9138_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9139_cis__neq__zero,axiom,
    ! [A3: real] :
      ( ( cis @ A3 )
     != zero_zero_complex ) ).

% cis_neq_zero
thf(fact_9140_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z3: complex] :
              ( ( power_power_complex @ Z3 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9141_set__decode__0,axiom,
    ! [X2: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).

% set_decode_0
thf(fact_9142_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_9143_set__decode__Suc,axiom,
    ! [N2: nat,X2: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X2 ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_9144_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_9145_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9146_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% set_decode_def
thf(fact_9147_forall__finite_I3_J,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ ( suc @ X2 ) ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X2 ) )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% forall_finite(3)
thf(fact_9148_forall__finite_I2_J,axiom,
    ! [P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ zero_zero_nat ) )
           => ( P @ I4 ) ) )
      = ( P @ zero_zero_nat ) ) ).

% forall_finite(2)
thf(fact_9149_Suc__0__mod__eq,axiom,
    ! [N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat
        @ ( N2
         != ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_0_mod_eq
thf(fact_9150_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_9151_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q4: int,R6: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R6 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_9152_forall__finite_I1_J,axiom,
    ! [P: nat > $o,I3: nat] :
      ( ( ord_less_nat @ I3 @ zero_zero_nat )
     => ( P @ I3 ) ) ).

% forall_finite(1)
thf(fact_9153_Comparator__Generator_OAll__less__Suc,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( suc @ X2 ) )
           => ( P @ I4 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ X2 )
           => ( P @ ( suc @ I4 ) ) ) ) ) ).

% Comparator_Generator.All_less_Suc
thf(fact_9154_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z3: complex] :
                ( ( power_power_complex @ Z3 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9155_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_9156_and__int_Oelims,axiom,
    ! [X2: int,Xa: int,Y2: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa )
        = Y2 )
     => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y2
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y2
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9157_real__root__zero,axiom,
    ! [N2: nat] :
      ( ( root @ N2 @ zero_zero_real )
      = zero_zero_real ) ).

% real_root_zero
thf(fact_9158_real__root__Suc__0,axiom,
    ! [X2: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
      = X2 ) ).

% real_root_Suc_0
thf(fact_9159_real__root__eq__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = ( root @ N2 @ Y2 ) )
        = ( X2 = Y2 ) ) ) ).

% real_root_eq_iff
thf(fact_9160_root__0,axiom,
    ! [X2: real] :
      ( ( root @ zero_zero_nat @ X2 )
      = zero_zero_real ) ).

% root_0
thf(fact_9161_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_9162_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_9163_real__root__eq__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = zero_zero_real )
        = ( X2 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9164_real__root__less__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ X2 @ Y2 ) ) ) ).

% real_root_less_iff
thf(fact_9165_real__root__le__iff,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ X2 @ Y2 ) ) ) ).

% real_root_le_iff
thf(fact_9166_real__root__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_9167_real__root__eq__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X2 )
          = one_one_real )
        = ( X2 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9168_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).

% real_root_gt_0_iff
thf(fact_9169_real__root__lt__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ zero_zero_real )
        = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9170_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).

% real_root_ge_0_iff
thf(fact_9171_real__root__le__0__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9172_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_real @ one_one_real @ Y2 ) ) ) ).

% real_root_gt_1_iff
thf(fact_9173_real__root__lt__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X2 ) @ one_one_real )
        = ( ord_less_real @ X2 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9174_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y2 ) )
        = ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).

% real_root_ge_1_iff
thf(fact_9175_real__root__le__1__iff,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ one_one_real )
        = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9176_and__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_9177_and__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_9178_real__root__pow__pos2,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos2
thf(fact_9179_and__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_9180_and__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_9181_real__root__pos__pos__le,axiom,
    ! [X2: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ).

% real_root_pos_pos_le
thf(fact_9182_AND__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) ) ) ).

% AND_lower
thf(fact_9183_AND__upper1,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ X2 ) ) ).

% AND_upper1
thf(fact_9184_AND__upper2,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Y2 ) ) ).

% AND_upper2
thf(fact_9185_AND__upper1_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_9186_AND__upper2_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_eq_int @ Y2 @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_9187_real__root__less__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X2 @ Y2 )
       => ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_less_mono
thf(fact_9188_real__root__le__mono,axiom,
    ! [N2: nat,X2: real,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X2 @ Y2 )
       => ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N2 @ Y2 ) ) ) ) ).

% real_root_le_mono
thf(fact_9189_real__root__power,axiom,
    ! [N2: nat,X2: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X2 @ K ) )
        = ( power_power_real @ ( root @ N2 @ X2 ) @ K ) ) ) ).

% real_root_power
thf(fact_9190_real__root__abs,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X2 ) )
        = ( abs_abs_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_abs
thf(fact_9191_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_9192_AND__upper1_H_H,axiom,
    ! [Y2: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y2 @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_9193_AND__upper2_H_H,axiom,
    ! [Y2: int,Z: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( ord_less_int @ Y2 @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_9194_real__root__gt__zero,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9195_real__root__strict__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N3 )
       => ( ( ord_less_real @ one_one_real @ X2 )
         => ( ord_less_real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9196_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9197_root__abs__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y2 @ N2 ) ) )
        = ( abs_abs_real @ Y2 ) ) ) ).

% root_abs_power
thf(fact_9198_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_9199_real__root__pos__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X2 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9200_real__root__strict__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N3 )
       => ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( ( ord_less_real @ X2 @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9201_real__root__decreasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N3 )
       => ( ( ord_less_eq_real @ one_one_real @ X2 )
         => ( ord_less_eq_real @ ( root @ N3 @ X2 ) @ ( root @ N2 @ X2 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9202_odd__real__root__pow,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
        = X2 ) ) ).

% odd_real_root_pow
thf(fact_9203_odd__real__root__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ( power_power_real @ Y2 @ N2 )
          = X2 )
       => ( ( root @ N2 @ X2 )
          = Y2 ) ) ) ).

% odd_real_root_unique
thf(fact_9204_odd__real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X2 @ N2 ) )
        = X2 ) ) ).

% odd_real_root_power_cancel
thf(fact_9205_real__root__pow__pos,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( power_power_real @ ( root @ N2 @ X2 ) @ N2 )
          = X2 ) ) ) ).

% real_root_pow_pos
thf(fact_9206_real__root__power__cancel,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
       => ( ( root @ N2 @ ( power_power_real @ X2 @ N2 ) )
          = X2 ) ) ) ).

% real_root_power_cancel
thf(fact_9207_real__root__pos__unique,axiom,
    ! [N2: nat,Y2: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
       => ( ( ( power_power_real @ Y2 @ N2 )
            = X2 )
         => ( ( root @ N2 @ X2 )
            = Y2 ) ) ) ) ).

% real_root_pos_unique
thf(fact_9208_real__root__increasing,axiom,
    ! [N2: nat,N3: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
         => ( ( ord_less_eq_real @ X2 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X2 ) @ ( root @ N3 @ X2 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9209_log__root,axiom,
    ! [N2: nat,A3: real,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A3 )
       => ( ( log @ B3 @ ( root @ N2 @ A3 ) )
          = ( divide_divide_real @ ( log @ B3 @ A3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_9210_log__base__root,axiom,
    ! [N2: nat,B3: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ( log @ ( root @ N2 @ B3 ) @ X2 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B3 @ X2 ) ) ) ) ) ).

% log_base_root
thf(fact_9211_ln__root,axiom,
    ! [N2: nat,B3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B3 )
       => ( ( ln_ln_real @ ( root @ N2 @ B3 ) )
          = ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_9212_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_9213_root__powr__inverse,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( ( root @ N2 @ X2 )
          = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9214_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L
            @ ( if_int
              @ ( L
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_9215_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_9216_and__int_Opelims,axiom,
    ! [X2: int,Xa: int,Y2: int] :
      ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y2
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y2
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_9217_and__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_9218_and__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_9219_and__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_9220_and__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_9221_and__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_Suc_0_eq
thf(fact_9222_Suc__0__and__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Suc_0_and_eq
thf(fact_9223_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat
          @ ( ( M3 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_9224_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_9225_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K2: int,L3: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L3 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_9226_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I2: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I2 @ J2 )
               => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
             => ( P @ I2 @ J2 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_9227_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_9228_take__bit__num__simps_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(2)
thf(fact_9229_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_9230_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_9231_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9232_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9233_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9234_take__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_Suc_0
thf(fact_9235_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_9236_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9237_take__bit__mult,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).

% take_bit_mult
thf(fact_9238_take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_9239_take__bit__diff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).

% take_bit_diff
thf(fact_9240_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_9241_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_9242_num__induct,axiom,
    ! [P: num > $o,X2: num] :
      ( ( P @ one )
     => ( ! [X3: num] :
            ( ( P @ X3 )
           => ( P @ ( inc @ X3 ) ) )
       => ( P @ X2 ) ) ) ).

% num_induct
thf(fact_9243_add__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( plus_plus_num @ X2 @ ( inc @ Y2 ) )
      = ( inc @ ( plus_plus_num @ X2 @ Y2 ) ) ) ).

% add_inc
thf(fact_9244_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_9245_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_9246_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_9247_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_9248_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_9249_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_9250_inc_Osimps_I3_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit1 @ X2 ) )
      = ( bit0 @ ( inc @ X2 ) ) ) ).

% inc.simps(3)
thf(fact_9251_inc_Osimps_I2_J,axiom,
    ! [X2: num] :
      ( ( inc @ ( bit0 @ X2 ) )
      = ( bit1 @ X2 ) ) ).

% inc.simps(2)
thf(fact_9252_add__One,axiom,
    ! [X2: num] :
      ( ( plus_plus_num @ X2 @ one )
      = ( inc @ X2 ) ) ).

% add_One
thf(fact_9253_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_9254_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_9255_mult__inc,axiom,
    ! [X2: num,Y2: num] :
      ( ( times_times_num @ X2 @ ( inc @ Y2 ) )
      = ( plus_plus_num @ ( times_times_num @ X2 @ Y2 ) @ X2 ) ) ).

% mult_inc
thf(fact_9256_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_9257_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_9258_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_9259_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_9260_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_nat_def
thf(fact_9261_take__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_9262_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_9263_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_def
thf(fact_9264_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_9265_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_9266_take__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_9267_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_9268_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_9269_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_9270_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_9271_take__bit__incr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_9272_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_9273_take__bit__int__less__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_9274_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_9275_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_9276_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_9277_uint32_Osize__eq,axiom,
    ( size_size_uint32
    = ( ^ [P5: uint32] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% uint32.size_eq
thf(fact_9278_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9279_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9280_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9281_mask__nat__positive__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% mask_nat_positive_iff
thf(fact_9282_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9283_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_9284_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_9285_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_9286_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9287_less__mask,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).

% less_mask
thf(fact_9288_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_9289_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9290_and__not__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9291_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_9292_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_9293_Suc__mask__eq__exp,axiom,
    ! [N2: nat] :
      ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_mask_eq_exp
thf(fact_9294_mask__nat__less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% mask_nat_less_exp
thf(fact_9295_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_9296_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_9297_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_9298_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N12: num] : ( some_num @ ( bit1 @ N12 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_9299_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_9300_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9301_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ N2 @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] : ( some_num @ one )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9302_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9303_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M3: num] :
          ( produc478579273971653890on_num
          @ ^ [A2: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X )
              @ A2 )
          @ ( product_Pair_nat_num @ N @ M3 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9304_powr__int,axiom,
    ! [X2: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X2 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9305_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9306_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9307_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9308_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9309_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9310_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9311_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_9312_nat__zminus__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9313_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_9314_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_9315_diff__nat__numeral,axiom,
    ! [V: num,V4: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V4 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V4 ) ) ) ) ).

% diff_nat_numeral
thf(fact_9316_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y2: int,X2: num,N2: nat] :
      ( ( ( nat2 @ Y2 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( Y2
        = ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9317_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,Y2: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 )
        = ( nat2 @ Y2 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 )
        = Y2 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9318_nat__ceiling__le__eq,axiom,
    ! [X2: real,A3: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A3 )
      = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A3 ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9319_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_9320_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_9321_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9322_numeral__power__less__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A3 ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9323_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) )
      = ( ord_less_eq_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9324_numeral__power__le__nat__cancel__iff,axiom,
    ! [X2: num,N2: nat,A3: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N2 ) @ ( nat2 @ A3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N2 ) @ A3 ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9325_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9326_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9327_nat__mono,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ X2 @ Y2 )
     => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).

% nat_mono
thf(fact_9328_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_9329_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P4 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_9330_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P4 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_9331_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_9332_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_9333_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_9334_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M3: nat,N: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_9335_nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% nat_mask_eq
thf(fact_9336_ln__real__def,axiom,
    ( ln_ln_real
    = ( ^ [X: real] :
          ( the_real
          @ ^ [U2: real] :
              ( ( exp_real @ U2 )
              = X ) ) ) ) ).

% ln_real_def
thf(fact_9337_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_9338_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_9339_nat__le__iff,axiom,
    ! [X2: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N2 )
      = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_9340_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_9341_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_9342_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9343_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M3: nat,N: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% and_nat_def
thf(fact_9344_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_9345_nat__minus__as__int,axiom,
    ( minus_minus_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_minus_as_int
thf(fact_9346_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_9347_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A2: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_9348_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_9349_ln__neg__is__const,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( ln_ln_real @ X2 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9350_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ M5 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9351_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M5: nat] : ( suc @ ( ord_max_nat @ N2 @ M5 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9352_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_9353_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_9354_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_9355_split__nat,axiom,
    ! [P: nat > $o,I: int] :
      ( ( P @ ( nat2 @ I ) )
      = ( ! [N: nat] :
            ( ( I
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P @ N ) )
        & ( ( ord_less_int @ I @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_9356_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_9357_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_9358_le__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% le_nat_iff
thf(fact_9359_Suc__as__int,axiom,
    ( suc
    = ( ^ [A2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9360_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_9361_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_9362_nat__diff__distrib_H,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( nat2 @ ( minus_minus_int @ X2 @ Y2 ) )
          = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9363_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_9364_nat__div__distrib_H,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y2 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib'
thf(fact_9365_nat__div__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( nat2 @ ( divide_divide_int @ X2 @ Y2 ) )
        = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ).

% nat_div_distrib
thf(fact_9366_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_9367_nat__mod__distrib,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y2 ) )
          = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_9368_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_9369_nat__floor__neg,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ X2 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9370_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_9371_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_9372_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_9373_floor__eq3,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_9374_le__nat__floor,axiom,
    ! [X2: nat,A3: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A3 )
     => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A3 ) ) ) ) ).

% le_nat_floor
thf(fact_9375_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9376_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9377_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_9378_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_9379_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_9380_nat__abs__int__diff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( ord_less_eq_nat @ A3 @ B3 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
          = ( minus_minus_nat @ B3 @ A3 ) ) )
      & ( ~ ( ord_less_eq_nat @ A3 @ B3 )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
          = ( minus_minus_nat @ A3 @ B3 ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9381_floor__eq4,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X2 )
     => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_9382_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_9383_arccos__def,axiom,
    ( arccos
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y ) ) ) ) ) ).

% arccos_def
thf(fact_9384_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_9385_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_9386_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9387_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9388_powr__real__of__int,axiom,
    ! [X2: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X2 @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9389_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y ) ) ) ) ) ).

% arcsin_def
thf(fact_9390_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9391_htt__vebt__inserti__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_inserti_invar_vebt
thf(fact_9392_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_9393_htt__vebt__buildupi_H,axiom,
    ! [N2: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% htt_vebt_buildupi'
thf(fact_9394_htt__vebt__buildupi,axiom,
    ! [N2: nat] : ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( vEBT_V441764108873111860ildupi @ N2 ) ) ).

% htt_vebt_buildupi
thf(fact_9395_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi'_univ
thf(fact_9396_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N2: nat] :
      ( ( U
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( time_htt_VEBT_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ U ) ) ) ).

% htt_vebt_buildupi_univ
thf(fact_9397_htt__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( time_htt_VEBT_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_inserti @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% htt_vebt_inserti
thf(fact_9398_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_9399_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_9400_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_9401_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_9402_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option4927543243414619207at_nat,Va2: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Vd: $o,Ve: $o] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ V @ Va2 @ Vb @ Vc ) @ ( vEBT_Leafi @ Vd @ Ve ) )
      = bot_bot_assn ) ).

% vebt_assn_raw.simps(3)
thf(fact_9403_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_9404_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9405_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B3: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9406_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B3: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9407_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_9408_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9409_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_9410_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_9411_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_9412_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_9413_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L ) ) )
          | ? [R6: int,L: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R6 ) )
              & ( ( sgn_sgn_int @ R6 )
                = ( sgn_sgn_int @ L ) )
              & ( ord_less_int @ ( abs_abs_int @ R6 ) @ ( abs_abs_int @ L ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R6 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9414_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q3 @ A22 ) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9415_divide__int__unfold,axiom,
    ! [L2: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9416_modulo__int__unfold,axiom,
    ! [L2: int,K: int,N2: nat,M: nat] :
      ( ( ( ( ( sgn_sgn_int @ L2 )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K )
            = zero_zero_int )
          | ( N2 = zero_zero_nat ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( sgn_sgn_int @ L2 )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K )
              = zero_zero_int )
            | ( N2 = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K )
              = ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L2 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L2 )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9417_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L: int] :
          ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9418_vebt__buildupi__rule,axiom,
    ! [N2: nat] : ( time_htt_VEBT_VEBTi @ ( pure_assn @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% vebt_buildupi_rule
thf(fact_9419_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z3: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9420_sgn__div__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( ( divide_divide_int @ A3 @ B3 )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( divide_divide_int @ A3 @ B3 ) )
        = ( sgn_sgn_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% sgn_div_eq_sgn_mult
thf(fact_9421_highi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R6: nat] :
          ( pure_assn
          @ ( R6
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) )
      @ one_one_nat ) ).

% highi_hT
thf(fact_9422_lowi__hT,axiom,
    ! [X2: nat,N2: nat] :
      ( time_htt_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R6: nat] :
          ( pure_assn
          @ ( R6
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) )
      @ one_one_nat ) ).

% lowi_hT
thf(fact_9423_vebt__minti__hT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R6: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_mint @ T2 ) ) ) )
      @ one_one_nat ) ).

% vebt_minti_hT
thf(fact_9424_vebt__maxti__hT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R6: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_maxt @ T2 ) ) ) )
      @ one_one_nat ) ).

% vebt_maxti_hT
thf(fact_9425_minNrulli__ruleT,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R6: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_VEBT_minNull @ T2 ) ) ) )
      @ one_one_nat ) ).

% minNrulli_ruleT
thf(fact_9426_zero__le__sgn__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).

% zero_le_sgn_iff
thf(fact_9427_sgn__le__0__iff,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_9428_htt__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( time_htt_o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
      @ ^ [R6: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
      @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_height @ T2 ) ) ) ) ).

% htt_vebt_memberi
thf(fact_9429_htt__vebt__memberi__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt_o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_memberi @ Ti @ X2 )
        @ ^ [R6: $o] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_member @ T2 @ X2 ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_memberi_invar_vebt
thf(fact_9430_htt__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( time_htt_option_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_succi @ Ti @ X2 )
        @ ^ [R6: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_succ @ T2 @ X2 ) ) ) )
        @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( nat2 @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ) ) ).

% htt_vebt_succi
thf(fact_9431_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A2: rat] : ( if_rat @ ( A2 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A2 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9432_obtain__pos__sum,axiom,
    ! [R: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T3: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T3 )
               => ( R
                 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9433_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_9434_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A2: rat] : ( if_rat @ ( ord_less_rat @ A2 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A2 ) @ A2 ) ) ) ).

% abs_rat_def
thf(fact_9435_sgn__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X2 ) )
        = ( sgn_sgn_real @ X2 ) ) ) ).

% sgn_root
thf(fact_9436_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( cis @ ( arg @ Z ) )
        = ( sgn_sgn_complex @ Z ) ) ) ).

% cis_Arg
thf(fact_9437_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A2: real] : ( if_real @ ( A2 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A2 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9438_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9439_sgn__power__injE,axiom,
    ! [A3: real,N2: nat,X2: real,B3: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A3 ) @ ( power_power_real @ ( abs_abs_real @ A3 ) @ N2 ) )
        = X2 )
     => ( ( X2
          = ( times_times_real @ ( sgn_sgn_real @ B3 ) @ ( power_power_real @ ( abs_abs_real @ B3 ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A3 = B3 ) ) ) ) ).

% sgn_power_injE
thf(fact_9440_root__sgn__power,axiom,
    ! [N2: nat,Y2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) ) )
        = Y2 ) ) ).

% root_sgn_power
thf(fact_9441_sgn__power__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X2 ) ) @ N2 ) )
        = X2 ) ) ).

% sgn_power_root
thf(fact_9442_cis__Arg__unique,axiom,
    ! [Z: complex,X2: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X2 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
       => ( ( ord_less_eq_real @ X2 @ pi )
         => ( ( arg @ Z )
            = X2 ) ) ) ) ).

% cis_Arg_unique
thf(fact_9443_split__root,axiom,
    ! [P: real > $o,N2: nat,X2: real] :
      ( ( P @ ( root @ N2 @ X2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
                = X2 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_9444_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_9445_arctan__inverse,axiom,
    ! [X2: real] :
      ( ( X2 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X2 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X2 ) ) ) ) ).

% arctan_inverse
thf(fact_9446_vebt__pred_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_predi @ T2 @ Ti @ X2 )
        @ ^ [R6: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_pred @ T2 @ X2 ) ) ) ) ) ) ).

% vebt_pred'_rf_abstr
thf(fact_9447_vebt__succi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,N2: nat,Ti: vEBT_VEBTi,X2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_vebt_succi @ T2 @ Ti @ X2 )
        @ ^ [R6: option_nat] :
            ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_succ @ T2 @ X2 ) ) ) ) ) ) ).

% vebt_succi'_rf_abstr
thf(fact_9448_vebt__memberi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V854960066525838166emberi @ T2 @ Ti @ X2 )
      @ ^ [R6: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_member @ T2 @ X2 ) ) ) ) ) ).

% vebt_memberi'_rf_abstr
thf(fact_9449_vebt__minti__h,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_minti @ Ti )
      @ ^ [R6: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_mint @ T2 ) ) ) ) ) ).

% vebt_minti_h
thf(fact_9450_vebt__maxti__h,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_7629718768684598413on_nat @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_vebt_maxti @ Ti )
      @ ^ [R6: option_nat] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_vebt_maxt @ T2 ) ) ) ) ) ).

% vebt_maxti_h
thf(fact_9451_minNulli__rule,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi] :
      ( hoare_hoare_triple_o @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_VEBT_minNulli @ Ti )
      @ ^ [R6: $o] :
          ( times_times_assn @ ( vEBT_vebt_assn_raw @ T2 @ Ti )
          @ ( pure_assn
            @ ( R6
              = ( vEBT_VEBT_minNull @ T2 ) ) ) ) ) ).

% minNulli_rule
thf(fact_9452_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_maxti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R6: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_maxtilist
thf(fact_9453_vebt__mintilist,axiom,
    ! [I: nat,Ts: list_VEBT_VEBT,Tsi: list_VEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ts ) )
     => ( hoare_7629718768684598413on_nat @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) @ ( vEBT_vebt_minti @ ( nth_VEBT_VEBTi @ Tsi @ I ) )
        @ ^ [R6: option_nat] :
            ( times_times_assn
            @ ( pure_assn
              @ ( R6
                = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ Ts @ I ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Ts @ Tsi ) ) ) ) ).

% vebt_mintilist
thf(fact_9454_vebt__inserti_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Ti: vEBT_VEBTi,X2: nat] : ( hoare_1429296392585015714_VEBTi @ ( vEBT_vebt_assn_raw @ T2 @ Ti ) @ ( vEBT_V3964819847710782039nserti @ T2 @ Ti @ X2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_insert @ T2 @ X2 ) ) ) ).

% vebt_inserti'_rf_abstr
thf(fact_9455_highi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_highi @ X2 @ N2 )
      @ ^ [R6: nat] :
          ( pure_assn
          @ ( R6
            = ( vEBT_VEBT_high @ X2 @ N2 ) ) ) ) ).

% highi_h
thf(fact_9456_lowi__h,axiom,
    ! [X2: nat,N2: nat] :
      ( hoare_3067605981109127869le_nat @ one_one_assn @ ( vEBT_VEBT_lowi @ X2 @ N2 )
      @ ^ [R6: nat] :
          ( pure_assn
          @ ( R6
            = ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).

% lowi_h
thf(fact_9457_builupi_Hcorr,axiom,
    ! [N2: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_V739175172307565963ildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) ) ).

% builupi'corr
thf(fact_9458_builupicorr,axiom,
    ! [N2: nat] : ( hoare_1429296392585015714_VEBTi @ one_one_assn @ ( vEBT_vebt_buildupi @ N2 ) @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_buildup @ N2 ) ) ) ).

% builupicorr
thf(fact_9459_Arg__def,axiom,
    ( arg
    = ( ^ [Z3: complex] :
          ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A2: real] :
                ( ( ( sgn_sgn_complex @ Z3 )
                  = ( cis @ A2 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A2 )
                & ( ord_less_eq_real @ A2 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9460_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H6: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H6 )
         => Phi )
     => ( hoare_8945653483474564448t_unit @ P @ ( refine_Imp_assert @ Phi )
        @ ^ [Uu: product_unit] : P ) ) ).

% assert'_rule
thf(fact_9461_vebt__assn__raw_Oelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y2: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
        = Y2 )
     => ( ! [A: $o,B: $o] :
            ( ( X2
              = ( vEBT_Leaf @ A @ B ) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xa
                  = ( vEBT_Leafi @ Ai @ Bi ) )
               => ( Y2
                 != ( pure_assn
                    @ ( ( Ai = A )
                      & ( Bi = B ) ) ) ) ) )
       => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) )
             => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                  ( ( Xa
                    = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                 => ( Y2
                   != ( times_times_assn
                      @ ( times_times_assn
                        @ ( pure_assn
                          @ ( ( Mmoi = Mmo )
                            & ( Degi = Deg ) ) )
                        @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi ) )
                      @ ( ex_ass463751140784270563_VEBTi
                        @ ^ [Tree_is: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is ) ) ) ) ) ) )
         => ( ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( X2
                  = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
             => ( ? [Vd3: $o,Ve3: $o] :
                    ( Xa
                    = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
               => ( Y2 != bot_bot_assn ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] :
                    ( X2
                    = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
               => ( ? [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                      ( Xa
                      = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ( Y2 != bot_bot_assn ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
thf(fact_9462_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option4927543243414619207at_nat,Deg4: nat,Tree_list2: list_VEBT_VEBT,Summary4: vEBT_VEBT,Mmoi2: option4927543243414619207at_nat,Degi2: nat,Tree_array2: array_VEBT_VEBTi,Summaryi2: vEBT_VEBTi] :
      ( ( vEBT_vebt_assn_raw @ ( vEBT_Node @ Mmo2 @ Deg4 @ Tree_list2 @ Summary4 ) @ ( vEBT_Nodei @ Mmoi2 @ Degi2 @ Tree_array2 @ Summaryi2 ) )
      = ( times_times_assn
        @ ( times_times_assn
          @ ( pure_assn
            @ ( ( Mmoi2 = Mmo2 )
              & ( Degi2 = Deg4 ) ) )
          @ ( vEBT_vebt_assn_raw @ Summary4 @ Summaryi2 ) )
        @ ( ex_ass463751140784270563_VEBTi
          @ ^ [Tree_is: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array2 @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list2 @ Tree_is ) ) ) ) ) ).

% vebt_assn_raw.simps(2)
thf(fact_9463_vebt__assn__raw_Opelims,axiom,
    ! [X2: vEBT_VEBT,Xa: vEBT_VEBTi,Y2: assn] :
      ( ( ( vEBT_vebt_assn_raw @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ X2 @ Xa ) )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xa
                    = ( vEBT_Leafi @ Ai @ Bi ) )
                 => ( ( Y2
                      = ( pure_assn
                        @ ( ( Ai = A )
                          & ( Bi = B ) ) ) )
                   => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ A @ B ) @ ( vEBT_Leafi @ Ai @ Bi ) ) ) ) ) )
         => ( ! [Mmo: option4927543243414619207at_nat,Deg: nat,Tree_list: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) )
               => ! [Mmoi: option4927543243414619207at_nat,Degi: nat,Tree_array: array_VEBT_VEBTi,Summaryi: vEBT_VEBTi] :
                    ( ( Xa
                      = ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) )
                   => ( ( Y2
                        = ( times_times_assn
                          @ ( times_times_assn
                            @ ( pure_assn
                              @ ( ( Mmoi = Mmo )
                                & ( Degi = Deg ) ) )
                            @ ( vEBT_vebt_assn_raw @ Summary @ Summaryi ) )
                          @ ( ex_ass463751140784270563_VEBTi
                            @ ^ [Tree_is: list_VEBT_VEBTi] : ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ Tree_array @ Tree_is ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ Tree_list @ Tree_is ) ) ) ) )
                     => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ Mmo @ Deg @ Tree_list @ Summary ) @ ( vEBT_Nodei @ Mmoi @ Degi @ Tree_array @ Summaryi ) ) ) ) ) )
           => ( ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: list_VEBT_VEBT,Vc3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xa
                        = ( vEBT_Leafi @ Vd3 @ Ve3 ) )
                     => ( ( Y2 = bot_bot_assn )
                       => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Node @ V2 @ Va3 @ Vb3 @ Vc3 ) @ ( vEBT_Leafi @ Vd3 @ Ve3 ) ) ) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( X2
                      = ( vEBT_Leaf @ Vd3 @ Ve3 ) )
                   => ! [V2: option4927543243414619207at_nat,Va3: nat,Vb3: array_VEBT_VEBTi,Vc3: vEBT_VEBTi] :
                        ( ( Xa
                          = ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) )
                       => ( ( Y2 = bot_bot_assn )
                         => ~ ( accp_P7675410724331315407_VEBTi @ vEBT_v8524038756793281170aw_rel @ ( produc6084888613844515218_VEBTi @ ( vEBT_Leaf @ Vd3 @ Ve3 ) @ ( vEBT_Nodei @ V2 @ Va3 @ Vb3 @ Vc3 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
thf(fact_9464_assnle,axiom,
    ! [TreeList2: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] : ( entails @ ( times_times_assn @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) ).

% assnle
thf(fact_9465_big__assn__simp,axiom,
    ! [H2: nat,TreeList2: list_VEBT_VEBT,L2: nat,X2: vEBT_VEBTi,Xaa: option_nat,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( entails
        @ ( times_times_assn
          @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) @ X2 )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) ) ) )
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) )
        @ ( times_times_assn
          @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) )
            @ ( pure_assn
              @ ( Xaa
                = ( vEBT_vebt_mint @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) ) ) )
          @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ).

% big_assn_simp
thf(fact_9466_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList2: list_VEBT_VEBT,Xaa: vEBT_VEBT,L2: nat,X2: vEBT_VEBTi,Xb: option_nat,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ H2 @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( Xaa
          = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
       => ( entails
          @ ( times_times_assn
            @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Xaa @ X2 )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ H2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) )
          @ ( times_times_assn
            @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) )
              @ ( pure_assn
                @ ( Xb
                  = ( vEBT_vebt_mint @ Xaa ) ) ) )
            @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Xaa ) @ ( list_u6098035379799741383_VEBTi @ Tree_is2 @ H2 @ X2 ) ) ) ) ) ) ).

% big_assn_simp'
thf(fact_9467_local_Oext,axiom,
    ! [Y2: nat,TreeList2: list_VEBT_VEBT,X13: array_VEBT_VEBTi,Tree_is2: list_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y2 @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( entails @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList2 @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList2 @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) ) ) ) ).

% local.ext
thf(fact_9468_recomp,axiom,
    ! [I: nat,TreeList2: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ I ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) ) ).

% recomp
thf(fact_9469_repack,axiom,
    ! [I: nat,TreeList2: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,Rest: assn,X13: array_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ I ) ) @ Rest ) @ ( times_times_assn @ ( times_times_assn @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ Rest @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) ) ).

% repack
thf(fact_9470_txe,axiom,
    ! [Y2: nat,TreeList2: list_VEBT_VEBT,Tree_is2: list_VEBT_VEBTi,X13: array_VEBT_VEBTi,Summary4: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( ( ord_less_nat @ Y2 @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( entails @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ ( nth_VEBT_VEBT @ TreeList2 @ Y2 ) @ ( nth_VEBT_VEBTi @ Tree_is2 @ Y2 ) ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) ) @ ( vEBT_L1528199826722428489_VEBTi @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) @ ( times_times_assn @ ( times_times_assn @ ( vEBT_vebt_assn_raw @ Summary4 @ X14 ) @ ( snga_assn_VEBT_VEBTi @ X13 @ Tree_is2 ) ) @ ( vEBT_L6296928887356842470_VEBTi @ vEBT_vebt_assn_raw @ TreeList2 @ Tree_is2 ) ) ) ) ).

% txe
thf(fact_9471_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
            & ( P @ M3 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9472_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N2 )
           => ( P @ M3 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9473_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9474_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9475_atLeast0__lessThan__Suc,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9476_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N3 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9477_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9478_prod__Suc__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_Suc_fact
thf(fact_9479_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9480_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_9481_atLeast1__lessThan__eq__remove0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9482_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_9483_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9484_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A3: nat > nat,B3: nat > nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A3 @ I2 ) @ ( A3 @ J2 ) ) ) )
     => ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A3 @ I4 ) @ ( B3 @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9485_root__def,axiom,
    ( root
    = ( ^ [N: nat,X: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9486_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : ( finite6017078050557962740nteger @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U ) ) ).

% finite_atLeastZeroLessThan_integer
thf(fact_9487_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9488_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_9489_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_9490_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_9491_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_9492_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_9493_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_9494_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_9495_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_9496_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_9497_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_9498_bin__nth__minus__Bit0,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ N2 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit0
thf(fact_9499_bin__nth__minus__Bit1,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ N2 )
        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% bin_nth_minus_Bit1
thf(fact_9500_bit__minus__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).

% bit_minus_int_iff
thf(fact_9501_bit__not__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff
thf(fact_9502_bit__and__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        & ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_and_int_iff
thf(fact_9503_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_9504_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_9505_bit__not__int__iff_H,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff'
thf(fact_9506_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_9507_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_9508_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_9509_and__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_9510_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_9511_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some_num @ Q2 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = ( numeral_numeral_int @ Q2 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9512_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N4: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N4 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).

% int_bit_bound
thf(fact_9513_and__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9514_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_9515_and__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_9516_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9517_and__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9518_and__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9519_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9520_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_9521_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_int_def
thf(fact_9522_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9523_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9524_Bit__Operations_Oset__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% Bit_Operations.set_bit_eq
thf(fact_9525_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% unset_bit_eq
thf(fact_9526_take__bit__Suc__from__most,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_Suc_from_most
thf(fact_9527_int__not__code_I1_J,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_not_code(1)
thf(fact_9528_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L )
          @ ( if_int
            @ ( L
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9529_num_Osize__gen_I3_J,axiom,
    ! [X34: num] :
      ( ( size_num @ ( bit1 @ X34 ) )
      = ( plus_plus_nat @ ( size_num @ X34 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_9530_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_9531_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_9532_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_9533_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_9534_or__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9535_or__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9536_or__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(2)
thf(fact_9537_or__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9538_or__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% or_nat_numerals(1)
thf(fact_9539_or__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9540_or__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9541_and__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9542_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).

% xor_int_def
thf(fact_9543_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_9544_OR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) ) ) ) ).

% OR_lower
thf(fact_9545_bit__or__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        | ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_or_int_iff
thf(fact_9546_bit__xor__int__iff,axiom,
    ! [K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
       != ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).

% bit_xor_int_iff
thf(fact_9547_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% or_nat_def
thf(fact_9548_int__or__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se1409905431419307370or_int @ zero_zero_int @ J )
      = J ) ).

% int_or_code(1)
thf(fact_9549_int__or__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se1409905431419307370or_int @ I @ zero_zero_int )
      = I ) ).

% int_or_code(2)
thf(fact_9550_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_9551_bit__Suc__0__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_Suc_0_iff
thf(fact_9552_XOR__lower,axiom,
    ! [X2: int,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y2 ) ) ) ) ).

% XOR_lower
thf(fact_9553_int__xor__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se6526347334894502574or_int @ I @ zero_zero_int )
      = I ) ).

% int_xor_code(2)
thf(fact_9554_int__xor__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ J )
      = J ) ).

% int_xor_code(1)
thf(fact_9555_plus__and__or,axiom,
    ! [X2: int,Y2: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X2 @ Y2 ) @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) )
      = ( plus_plus_int @ X2 @ Y2 ) ) ).

% plus_and_or
thf(fact_9556_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).

% or_int_def
thf(fact_9557_not__bit__Suc__0__numeral,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).

% not_bit_Suc_0_numeral
thf(fact_9558_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_9559_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_9560_or__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9561_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_9562_or__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9563_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_9564_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_9565_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M3: nat,N: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_nat_def
thf(fact_9566_or__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9567_XOR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_9568_OR__upper,axiom,
    ! [X2: int,N2: nat,Y2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X2 )
     => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_9569_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9570_test__bit__int__code_I1_J,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ zero_zero_int @ N2 ) ).

% test_bit_int_code(1)
thf(fact_9571_int__and__code_I1_J,axiom,
    ! [J: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ J )
      = zero_zero_int ) ).

% int_and_code(1)
thf(fact_9572_int__and__code_I2_J,axiom,
    ! [I: int] :
      ( ( bit_se725231765392027082nd_int @ I @ zero_zero_int )
      = zero_zero_int ) ).

% int_and_code(2)
thf(fact_9573_or__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% or_Suc_0_eq
thf(fact_9574_Suc__0__or__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_or_eq
thf(fact_9575_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9576_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9577_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9578_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9579_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9580_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9581_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9582_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_9583_or__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9584_or__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9585_xor__nat__numerals_I4_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_9586_xor__nat__numerals_I3_J,axiom,
    ! [X2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_9587_xor__nat__numerals_I2_J,axiom,
    ! [Y2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9588_xor__nat__numerals_I1_J,axiom,
    ! [Y2: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y2 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y2 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9589_or__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9590_or__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9591_or__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9592_or__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9593_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9594_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9595_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9596_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9597_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_9598_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9599_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9600_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% xor_nat_def
thf(fact_9601_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_9602_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9603_or__not__num__neg_Oelims,axiom,
    ! [X2: num,Xa: num,Y2: num] :
      ( ( ( bit_or_not_num_neg @ X2 @ Xa )
        = Y2 )
     => ( ( ( X2 = one )
         => ( ( Xa = one )
           => ( Y2 != one ) ) )
       => ( ( ( X2 = one )
           => ! [M4: num] :
                ( ( Xa
                  = ( bit0 @ M4 ) )
               => ( Y2
                 != ( bit1 @ M4 ) ) ) )
         => ( ( ( X2 = one )
             => ! [M4: num] :
                  ( ( Xa
                    = ( bit1 @ M4 ) )
                 => ( Y2
                   != ( bit1 @ M4 ) ) ) )
           => ( ( ? [N4: num] :
                    ( X2
                    = ( bit0 @ N4 ) )
               => ( ( Xa = one )
                 => ( Y2
                   != ( bit0 @ one ) ) ) )
             => ( ! [N4: num] :
                    ( ( X2
                      = ( bit0 @ N4 ) )
                   => ! [M4: num] :
                        ( ( Xa
                          = ( bit0 @ M4 ) )
                       => ( Y2
                         != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
               => ( ! [N4: num] :
                      ( ( X2
                        = ( bit0 @ N4 ) )
                     => ! [M4: num] :
                          ( ( Xa
                            = ( bit1 @ M4 ) )
                         => ( Y2
                           != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
                 => ( ( ? [N4: num] :
                          ( X2
                          = ( bit1 @ N4 ) )
                     => ( ( Xa = one )
                       => ( Y2 != one ) ) )
                   => ( ! [N4: num] :
                          ( ( X2
                            = ( bit1 @ N4 ) )
                         => ! [M4: num] :
                              ( ( Xa
                                = ( bit0 @ M4 ) )
                             => ( Y2
                               != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
                     => ~ ! [N4: num] :
                            ( ( X2
                              = ( bit1 @ N4 ) )
                           => ! [M4: num] :
                                ( ( Xa
                                  = ( bit1 @ M4 ) )
                               => ( Y2
                                 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9604_numeral__or__not__num__eq,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9605_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9606_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9607_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N @ ( if_nat @ ( N = 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9608_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M3: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9609_Suc__0__xor__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% Suc_0_xor_eq
thf(fact_9610_xor__Suc__0__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% xor_Suc_0_eq
thf(fact_9611_int__lsb__numeral_I1_J,axiom,
    ~ ( least_4859182151741483524sb_int @ zero_zero_int ) ).

% int_lsb_numeral(1)
thf(fact_9612_int__lsb__numeral_I2_J,axiom,
    least_4859182151741483524sb_int @ one_one_int ).

% int_lsb_numeral(2)
thf(fact_9613_int__lsb__numeral_I6_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ).

% int_lsb_numeral(6)
thf(fact_9614_int__lsb__numeral_I3_J,axiom,
    least_4859182151741483524sb_int @ ( numeral_numeral_int @ one ) ).

% int_lsb_numeral(3)
thf(fact_9615_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ).

% int_lsb_numeral(7)
thf(fact_9616_int__lsb__numeral_I4_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ one_one_int ) ).

% int_lsb_numeral(4)
thf(fact_9617_int__lsb__numeral_I8_J,axiom,
    ! [W: num] :
      ~ ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) ) ).

% int_lsb_numeral(8)
thf(fact_9618_int__lsb__numeral_I5_J,axiom,
    least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) ).

% int_lsb_numeral(5)
thf(fact_9619_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : ( least_4859182151741483524sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) ) ).

% int_lsb_numeral(9)
thf(fact_9620_lsb__int__def,axiom,
    ( least_4859182151741483524sb_int
    = ( ^ [I4: int] : ( bit_se1146084159140164899it_int @ I4 @ zero_zero_nat ) ) ) ).

% lsb_int_def
thf(fact_9621_bin__last__conv__lsb,axiom,
    ( ( ^ [A2: int] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
    = least_4859182151741483524sb_int ) ).

% bin_last_conv_lsb
thf(fact_9622_pow_Osimps_I1_J,axiom,
    ! [X2: num] :
      ( ( pow @ X2 @ one )
      = X2 ) ).

% pow.simps(1)
thf(fact_9623_cis__multiple__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_9624_sin__times__pi__eq__0,axiom,
    ! [X2: real] :
      ( ( ( sin_real @ ( times_times_real @ X2 @ pi ) )
        = zero_zero_real )
      = ( member_real @ X2 @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_9625_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_9626_cos__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_9627_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,B2: int] : ( if_Pro3027730157355071871nt_int @ ( A2 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A2 ) @ B2 ) @ ( abs_abs_int @ A2 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9628_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_9629_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9630_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9631_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_9632_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_9633_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( ( quotient_of @ R )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% quotient_of_denom_pos
thf(fact_9634_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_9635_rat__uminus__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_uminus_code
thf(fact_9636_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D3: int] : ( ord_less_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9637_rat__floor__code,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [P5: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P5 ) ) ) ) ).

% rat_floor_code
thf(fact_9638_rat__abs__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int] : ( product_Pair_int_int @ ( abs_abs_int @ A2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_abs_code
thf(fact_9639_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q4: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A2: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B2: int,D3: int] : ( ord_less_eq_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ C3 @ B2 ) )
              @ ( quotient_of @ Q4 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9640_rat__sgn__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9641_quotient__of__int,axiom,
    ! [A3: int] :
      ( ( quotient_of @ ( of_int @ A3 ) )
      = ( product_Pair_int_int @ A3 @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9642_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9643_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ B2 @ C3 ) ) @ ( times_times_int @ C3 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9644_normalize__denom__zero,axiom,
    ! [P2: int] :
      ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9645_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( ( ord_less_int @ Q2 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).

% normalize_negative
thf(fact_9646_normalize__denom__pos,axiom,
    ! [R: product_prod_int_int,P2: int,Q2: int] :
      ( ( ( normalize @ R )
        = ( product_Pair_int_int @ P2 @ Q2 ) )
     => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).

% normalize_denom_pos
thf(fact_9647_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
         => ( ( times_times_int @ P2 @ S )
            = ( times_times_int @ R @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9648_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ D3 ) @ ( times_times_int @ C3 @ B2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9649_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A2: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B2: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ C3 @ D3 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9650_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9651_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_9652_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9653_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
      = ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9654_gcd__neg__numeral__2__int,axiom,
    ! [X2: int,N2: num] :
      ( ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( gcd_gcd_int @ X2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9655_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X2: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X2 )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X2 ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9656_gcd__0__left__int,axiom,
    ! [X2: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X2 )
      = ( abs_abs_int @ X2 ) ) ).

% gcd_0_left_int
thf(fact_9657_gcd__0__int,axiom,
    ! [X2: int] :
      ( ( gcd_gcd_int @ X2 @ zero_zero_int )
      = ( abs_abs_int @ X2 ) ) ).

% gcd_0_int
thf(fact_9658_gcd__ge__0__int,axiom,
    ! [X2: int,Y2: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y2 ) ) ).

% gcd_ge_0_int
thf(fact_9659_gcd__le1__int,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_int @ zero_zero_int @ A3 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A3 @ B3 ) @ A3 ) ) ).

% gcd_le1_int
thf(fact_9660_gcd__le2__int,axiom,
    ! [B3: int,A3: int] :
      ( ( ord_less_int @ zero_zero_int @ B3 )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A3 @ B3 ) @ B3 ) ) ).

% gcd_le2_int
thf(fact_9661_gcd__cases__int,axiom,
    ! [X2: int,Y2: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
         => ( P @ ( gcd_gcd_int @ X2 @ Y2 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
         => ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y2 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y2 ) ) ) )
         => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y2 @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y2 ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X2 @ Y2 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9662_gcd__non__0__int,axiom,
    ! [Y2: int,X2: int] :
      ( ( ord_less_int @ zero_zero_int @ Y2 )
     => ( ( gcd_gcd_int @ X2 @ Y2 )
        = ( gcd_gcd_int @ Y2 @ ( modulo_modulo_int @ X2 @ Y2 ) ) ) ) ).

% gcd_non_0_int
thf(fact_9663_gcd__unique__int,axiom,
    ! [D: int,A3: int,B3: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A3 )
        & ( dvd_dvd_int @ D @ B3 )
        & ! [E4: int] :
            ( ( ( dvd_dvd_int @ E4 @ A3 )
              & ( dvd_dvd_int @ E4 @ B3 ) )
           => ( dvd_dvd_int @ E4 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A3 @ B3 ) ) ) ).

% gcd_unique_int
thf(fact_9664_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K3: int,L: int] : ( abs_abs_int @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_9665_Frct__code__post_I1_J,axiom,
    ! [A3: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A3 ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9666_Frct__code__post_I2_J,axiom,
    ! [A3: int] :
      ( ( frct @ ( product_Pair_int_int @ A3 @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9667_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_9668_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_9669_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_9670_VEBT__internal_Ospace_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit0 @ one ) ) ) @ ( vEBT_VEBT_space @ Summary ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel2 @ ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) ) ) ) ) ) ) ).

% VEBT_internal.space.pelims
thf(fact_9671_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ( gcd_gcd_nat @ A3 @ B3 )
        = zero_zero_nat )
      = ( ( A3 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9672_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A3 )
      = A3 ) ).

% gcd_nat.left_neutral
thf(fact_9673_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A3 @ B3 ) )
      = ( ( A3 = zero_zero_nat )
        & ( B3 = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9674_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] :
      ( ( gcd_gcd_nat @ A3 @ zero_zero_nat )
      = A3 ) ).

% gcd_nat.right_neutral
thf(fact_9675_gcd__0__nat,axiom,
    ! [X2: nat] :
      ( ( gcd_gcd_nat @ X2 @ zero_zero_nat )
      = X2 ) ).

% gcd_0_nat
thf(fact_9676_gcd__0__left__nat,axiom,
    ! [X2: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X2 )
      = X2 ) ).

% gcd_0_left_nat
thf(fact_9677_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9678_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9679_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9680_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9681_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9682_gcd__le2__nat,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ B3 ) ) ).

% gcd_le2_nat
thf(fact_9683_gcd__le1__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A3 ) ) ).

% gcd_le1_nat
thf(fact_9684_gcd__non__0__nat,axiom,
    ! [Y2: nat,X2: nat] :
      ( ( Y2 != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X2 @ Y2 )
        = ( gcd_gcd_nat @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9685_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9686_gcd__nat_Oelims,axiom,
    ! [X2: nat,Xa: nat,Y2: nat] :
      ( ( ( gcd_gcd_nat @ X2 @ Xa )
        = Y2 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y2 = X2 ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y2
            = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9687_bezout__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero_nat )
     => ? [X3: nat,Y3: nat] :
          ( ( times_times_nat @ A3 @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y3 ) @ ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).

% bezout_nat
thf(fact_9688_bezout__gcd__nat_H,axiom,
    ! [B3: nat,A3: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ Y3 ) @ ( times_times_nat @ A3 @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A3 @ X3 ) @ ( times_times_nat @ B3 @ Y3 ) )
          = ( gcd_gcd_nat @ A3 @ B3 ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ Y3 ) @ ( times_times_nat @ B3 @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B3 @ X3 ) @ ( times_times_nat @ A3 @ Y3 ) )
          = ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9689_gcd__code__integer,axiom,
    ( gcd_gcd_Code_integer
    = ( ^ [K3: code_integer,L: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ) ) ).

% gcd_code_integer
thf(fact_9690_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_space2 @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( vEBT_VEBT_space2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_space2 @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_space_rel @ ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) ) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
thf(fact_9691_VEBT__internal_Ocnt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: real] :
      ( ( ( vEBT_VEBT_cnt @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2 = one_one_real )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( plus_plus_real @ ( plus_plus_real @ one_one_real @ ( vEBT_VEBT_cnt @ Summary ) ) @ ( foldr_real_real @ plus_plus_real @ ( map_VEBT_VEBT_real @ vEBT_VEBT_cnt @ TreeList3 ) @ zero_zero_real ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel2 @ ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) ) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
thf(fact_9692_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_cnt2 @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2 = one_one_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( plus_plus_nat @ ( plus_plus_nat @ one_one_nat @ ( vEBT_VEBT_cnt2 @ Summary ) ) @ ( foldr_nat_nat @ plus_plus_nat @ ( map_VEBT_VEBT_nat @ vEBT_VEBT_cnt2 @ TreeList3 ) @ zero_zero_nat ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_cnt_rel @ ( vEBT_Node @ Info @ Deg @ TreeList3 @ Summary ) ) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
thf(fact_9693_vebt__maxt_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( B
                   => ( Y2
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B
                   => ( ( A
                       => ( Y2
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9694_vebt__mint_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: option_nat] :
      ( ( ( vEBT_vebt_mint @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( ( A
                   => ( Y2
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A
                   => ( ( B
                       => ( Y2
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B
                       => ( Y2 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9695_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_t @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ A @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_i_n_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
thf(fact_9696_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_a_x_t @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2
                  = ( plus_plus_nat @ one_one_nat @ ( if_nat @ B @ one_one_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: list_VEBT_VEBT,Uz3: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T_m_a_x_t_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
thf(fact_9697_gcd__nat_Opelims,axiom,
    ! [X2: nat,Xa: nat,Y2: nat] :
      ( ( ( gcd_gcd_nat @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y2 = X2 ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y2
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X2 @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9698_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_T_m_i_n_N_u_l_l @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( ( Y2 = one_one_nat )
             => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ( Y2 = one_one_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu3 @ $true ) )
                 => ( ( Y2 = one_one_nat )
                   => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
                   => ( ( Y2 = one_one_nat )
                     => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) ) ) )
               => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                     => ( ( Y2 = one_one_nat )
                       => ~ ( accp_VEBT_VEBT @ vEBT_T5462971552011256508_l_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
thf(fact_9699_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Y2: $o] :
      ( ( ( vEBT_VEBT_minNull @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y2
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y2
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu3: $o] :
                  ( ( X2
                    = ( vEBT_Leaf @ Uu3 @ $true ) )
                 => ( ~ Y2
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                    ( ( X2
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
                   => ( Y2
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) ) ) )
               => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X2
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                     => ( ~ Y2
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9700_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ! [Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu3: $o] :
                ( ( X2
                  = ( vEBT_Leaf @ Uu3 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu3 @ $true ) ) )
           => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X2
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9701_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X2 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux3: list_VEBT_VEBT,Uy3: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9702_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list_VEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ S @ M )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
           => ( vEBT_invar_vebt @ X3 @ N2 ) )
       => ( ( M
            = ( suc @ N2 ) )
         => ( ! [X3: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Listy ) )
               => ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ X3 ) )
                  = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
           => ( ( ( semiri1314217659103216013at_int @ ( vEBT_VEBT_height @ S ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) )
             => ( ( semiri1314217659103216013at_int @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ S @ ( set_VEBT_VEBT2 @ Listy ) ) ) ) )
                = ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ) ) ) ) ).

% setceilmax
thf(fact_9703_height__compose__list,axiom,
    ! [T2: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ T2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ T2 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary4 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ).

% height_compose_list
thf(fact_9704_max__ins__scaled,axiom,
    ! [N2: nat,X14: vEBT_VEBT,M: nat,X13: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( vEBT_VEBT_height @ X14 ) ) @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ ( lattic8265883725875713057ax_nat @ ( insert_nat @ ( vEBT_VEBT_height @ X14 ) @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ).

% max_ins_scaled
thf(fact_9705_height__i__max,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,Foo: nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) @ ( ord_max_nat @ Foo @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ).

% height_i_max
thf(fact_9706_max__idx__list,axiom,
    ! [I: nat,X13: list_VEBT_VEBT,N2: nat,X14: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ X13 ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( vEBT_VEBT_height @ ( nth_VEBT_VEBT @ X13 @ I ) ) ) @ ( suc @ ( suc @ ( times_times_nat @ N2 @ ( ord_max_nat @ ( vEBT_VEBT_height @ X14 ) @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( set_VEBT_VEBT2 @ X13 ) ) ) ) ) ) ) ) ) ).

% max_idx_list
thf(fact_9707_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9708_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT] :
      ( ( vEBT_VEBT_height @ ( vEBT_Node @ Uu2 @ Deg4 @ TreeList2 @ Summary4 ) )
      = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary4 @ ( set_VEBT_VEBT2 @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.height.simps(2)
thf(fact_9709_VEBT__internal_Oheight_Oelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( ? [A: $o,B: $o] :
              ( X2
              = ( vEBT_Leaf @ A @ B ) )
         => ( Y2 != zero_zero_nat ) )
       => ~ ! [Uu3: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Uu3 @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
               != ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.height.elims
thf(fact_9710_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M3: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M3 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9711_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D3: nat] :
                ( ( dvd_dvd_nat @ D3 @ M )
                & ( dvd_dvd_nat @ D3 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9712_VEBT__internal_Oheight_Opelims,axiom,
    ! [X2: vEBT_VEBT,Y2: nat] :
      ( ( ( vEBT_VEBT_height @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leaf @ A @ B ) )
             => ( ( Y2 = zero_zero_nat )
               => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
         => ~ ! [Uu3: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Uu3 @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( plus_plus_nat @ one_one_nat @ ( lattic8265883725875713057ax_nat @ ( image_VEBT_VEBT_nat @ vEBT_VEBT_height @ ( insert_VEBT_VEBT @ Summary @ ( set_VEBT_VEBT2 @ TreeList3 ) ) ) ) ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_VEBT_height_rel @ ( vEBT_Node @ Uu3 @ Deg @ TreeList3 @ Summary ) ) ) ) ) ) ) ).

% VEBT_internal.height.pelims
thf(fact_9713_bij__betw__Suc,axiom,
    ! [M8: set_nat,N3: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M8 @ N3 )
      = ( ( image_nat_nat @ suc @ M8 )
        = N3 ) ) ).

% bij_betw_Suc
thf(fact_9714_range__mult,axiom,
    ! [A3: real] :
      ( ( ( A3 = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A3 ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A3 != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A3 ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9715_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D3: int] : ( dvd_dvd_int @ D3 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9716_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_9717_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D3: int] :
                ( ( dvd_dvd_int @ D3 @ M )
                & ( dvd_dvd_int @ D3 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9718_image__Suc__lessThan,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).

% image_Suc_lessThan
thf(fact_9719_image__Suc__atMost,axiom,
    ! [N2: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).

% image_Suc_atMost
thf(fact_9720_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9721_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9722_lessThan__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9723_atMost__Suc__eq__insert__0,axiom,
    ! [N2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9724_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9725_image__add__int__atLeastLessThan,axiom,
    ! [L2: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L2 )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
      = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9726_image__add__integer__atLeastLessThan,axiom,
    ! [L2: code_integer,U: code_integer] :
      ( ( image_4470545334726330049nteger
        @ ^ [X: code_integer] : ( plus_p5714425477246183910nteger @ X @ L2 )
        @ ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ ( minus_8373710615458151222nteger @ U @ L2 ) ) )
      = ( set_or8404916559141939852nteger @ L2 @ U ) ) ).

% image_add_integer_atLeastLessThan
thf(fact_9727_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_9728_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y2: nat,X2: nat] :
      ( ( ( ord_less_nat @ C @ Y2 )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y2 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y2 )
       => ( ( ( ord_less_nat @ X2 @ Y2 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X2 @ Y2 )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X2 @ Y2 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9729_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_9730_wait__rule,axiom,
    ! [N2: nat] :
      ( hoare_8945653483474564448t_unit @ one_one_assn @ ( heap_Time_wait @ N2 )
      @ ^ [Uu: product_unit] : one_one_assn ) ).

% wait_rule
thf(fact_9731_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M3: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M3 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9732_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9733_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9734_numeral__num__of__nat,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
        = N2 ) ) ).

% numeral_num_of_nat
thf(fact_9735_num__of__nat__One,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ one_one_nat )
     => ( ( num_of_nat @ N2 )
        = one ) ) ).

% num_of_nat_One
thf(fact_9736_num__of__nat__double,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
        = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).

% num_of_nat_double
thf(fact_9737_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9738_num__of__nat_Osimps_I2_J,axiom,
    ! [N2: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = ( inc @ ( num_of_nat @ N2 ) ) ) )
      & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( suc @ N2 ) )
          = one ) ) ) ).

% num_of_nat.simps(2)
thf(fact_9739_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9740_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9741_drop__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9742_drop__bit__Suc__minus__bit0,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9743_drop__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% drop_bit_of_Suc_0
thf(fact_9744_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_9745_drop__bit__Suc__minus__bit1,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9746_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_9747_drop__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se8568078237143864401it_int @ zero_zero_nat @ I )
      = I ) ).

% drop_bit_int_code(1)
thf(fact_9748_drop__bit__int__code_I2_J,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ zero_zero_int )
      = zero_zero_int ) ).

% drop_bit_int_code(2)
thf(fact_9749_drop__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9750_bin__rest__code,axiom,
    ! [I: int] :
      ( ( divide_divide_int @ I @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se8568078237143864401it_int @ one_one_nat @ I ) ) ).

% bin_rest_code
thf(fact_9751_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_int_def
thf(fact_9752_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9753_shiftr__integer__conv__div__pow2,axiom,
    ( bit_se3928097537394005634nteger
    = ( ^ [N: nat,X: code_integer] : ( divide6298287555418463151nteger @ X @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% shiftr_integer_conv_div_pow2
thf(fact_9754_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_9755_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9756_push__bit__of__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_Suc_0
thf(fact_9757_drop__bit__push__bit__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9758_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M3: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_9759_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M3: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9760_push__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_9761_push__bit__int__code_I1_J,axiom,
    ! [I: int] :
      ( ( bit_se545348938243370406it_int @ zero_zero_nat @ I )
      = I ) ).

% push_bit_int_code(1)
thf(fact_9762_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9763_shiftl__integer__conv__mult__pow2,axiom,
    ( bit_se7788150548672797655nteger
    = ( ^ [N: nat,X: code_integer] : ( times_3573771949741848930nteger @ X @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% shiftl_integer_conv_mult_pow2
thf(fact_9764_Bit__Operations_Oset__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% Bit_Operations.set_bit_int_def
thf(fact_9765_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9766_lsb__integer__code,axiom,
    ( least_7544222001954398261nteger
    = ( ^ [X: code_integer] : ( bit_se9216721137139052372nteger @ X @ zero_zero_nat ) ) ) ).

% lsb_integer_code
thf(fact_9767_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_9768_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_9769_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_int_def
thf(fact_9770_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_nat_def
thf(fact_9771_push__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% push_bit_minus_one
thf(fact_9772_Bit__integer_Oabs__eq,axiom,
    ! [Xa: int,X2: $o] :
      ( ( bits_Bit_integer @ ( code_integer_of_int @ Xa ) @ X2 )
      = ( code_integer_of_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ X2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ).

% Bit_integer.abs_eq
thf(fact_9773_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $false )
      = ( bit_se7788150548672797655nteger @ one_one_nat @ I ) ) ).

% Bit_integer_code(1)
thf(fact_9774_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] :
      ( ( bits_Bit_integer @ I @ $true )
      = ( plus_p5714425477246183910nteger @ ( bit_se7788150548672797655nteger @ one_one_nat @ I ) @ one_one_Code_integer ) ) ).

% Bit_integer_code(2)
thf(fact_9775_dup__1,axiom,
    ( ( code_dup @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dup_1
thf(fact_9776_bin__last__integer__nbe,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I4: code_integer] :
          ( ( modulo364778990260209775nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_nbe
thf(fact_9777_Code__Numeral_Odup__code_I1_J,axiom,
    ( ( code_dup @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% Code_Numeral.dup_code(1)
thf(fact_9778_bin__last__integer__code,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [I4: code_integer] :
          ( ( bit_se3949692690581998587nteger @ I4 @ one_one_Code_integer )
         != zero_z3403309356797280102nteger ) ) ) ).

% bin_last_integer_code
thf(fact_9779_bin__last__integer_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( bits_b8758750999018896077nteger @ ( code_integer_of_int @ X2 ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 ) ) ) ).

% bin_last_integer.abs_eq
thf(fact_9780_bitAND__integer__unfold,axiom,
    ( bit_se3949692690581998587nteger
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer
            @ ( X
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ Y
            @ ( bits_Bit_integer @ ( bit_se3949692690581998587nteger @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( bits_b8758750999018896077nteger @ X )
                & ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitAND_integer_unfold
thf(fact_9781_bitOR__integer__unfold,axiom,
    ( bit_se1080825931792720795nteger
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ Y
          @ ( if_Code_integer
            @ ( X
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( uminus1351360451143612070nteger @ one_one_Code_integer )
            @ ( bits_Bit_integer @ ( bit_se1080825931792720795nteger @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( bits_b8758750999018896077nteger @ X )
                | ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitOR_integer_unfold
thf(fact_9782_bin__rest__integer__code,axiom,
    ( bits_b2549910563261871055nteger
    = ( ^ [I4: code_integer] : ( divide6298287555418463151nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer_code
thf(fact_9783_bin__rest__integer_Oabs__eq,axiom,
    ! [X2: int] :
      ( ( bits_b2549910563261871055nteger @ ( code_integer_of_int @ X2 ) )
      = ( code_integer_of_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% bin_rest_integer.abs_eq
thf(fact_9784_bitXOR__integer__unfold,axiom,
    ( bit_se3222712562003087583nteger
    = ( ^ [X: code_integer,Y: code_integer] :
          ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ Y
          @ ( if_Code_integer
            @ ( X
              = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
            @ ( bit_ri7632146776885996613nteger @ Y )
            @ ( bits_Bit_integer @ ( bit_se3222712562003087583nteger @ ( bits_b2549910563261871055nteger @ X ) @ ( bits_b2549910563261871055nteger @ Y ) )
              @ ( ( ~ ( bits_b8758750999018896077nteger @ X ) )
                = ( bits_b8758750999018896077nteger @ Y ) ) ) ) ) ) ) ).

% bitXOR_integer_unfold
thf(fact_9785_Uint32__code,axiom,
    ( uint322
    = ( ^ [I4: code_integer] : ( if_uint32 @ ( bit_se9216721137139052372nteger @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uint32_signed @ ( minus_8373710615458151222nteger @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( uint32_signed @ ( bit_se3949692690581998587nteger @ I4 @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Uint32_code
thf(fact_9786_Uint32__signed__def,axiom,
    ( uint32_signed
    = ( ^ [I4: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ I4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I4 ) )
          @ ( undefi2040150642751712519uint32 @ uint322 @ I4 )
          @ ( uint322 @ I4 ) ) ) ) ).

% Uint32_signed_def
thf(fact_9787_vebt__minti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_minti @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A @ B ) )
             => ( ( ( A
                   => ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                  & ( ~ A
                   => ( ( B
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                      & ( ~ B
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Leafi @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: array_VEBT_VEBTi,Uz3: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Mi2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_minti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% vebt_minti.pelims
thf(fact_9788_vebt__maxti_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_T2636463487746394924on_nat] :
      ( ( ( vEBT_vebt_maxti @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ X2 )
       => ( ! [A: $o,B: $o] :
              ( ( X2
                = ( vEBT_Leafi @ A @ B ) )
             => ( ( ( B
                   => ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) ) )
                  & ( ~ B
                   => ( ( A
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) ) )
                      & ( ~ A
                       => ( Y2
                          = ( heap_T3487192422709364219on_nat @ none_nat ) ) ) ) ) )
               => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Leafi @ A @ B ) ) ) )
         => ( ! [Uu3: nat,Uv2: array_VEBT_VEBTi,Uw2: vEBT_VEBTi] :
                ( ( X2
                  = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) )
               => ( ( Y2
                    = ( heap_T3487192422709364219on_nat @ none_nat ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uu3 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux3: nat,Uy3: array_VEBT_VEBTi,Uz3: vEBT_VEBTi] :
                  ( ( X2
                    = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) )
                 => ( ( Y2
                      = ( heap_T3487192422709364219on_nat @ ( some_nat @ Ma2 ) ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_vebt_maxti_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux3 @ Uy3 @ Uz3 ) ) ) ) ) ) ) ) ).

% vebt_maxti.pelims
thf(fact_9789_VEBT__internal_OminNulli_Opelims,axiom,
    ! [X2: vEBT_VEBTi,Y2: heap_Time_Heap_o] :
      ( ( ( vEBT_VEBT_minNulli @ X2 )
        = Y2 )
     => ( ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ X2 )
       => ( ( ( X2
              = ( vEBT_Leafi @ $false @ $false ) )
           => ( ( Y2
                = ( heap_Time_return_o @ $true ) )
             => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X2
                  = ( vEBT_Leafi @ $true @ Uv2 ) )
               => ( ( Y2
                    = ( heap_Time_return_o @ $false ) )
                 => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ $true @ Uv2 ) ) ) )
           => ( ! [Uu3: $o] :
                  ( ( X2
                    = ( vEBT_Leafi @ Uu3 @ $true ) )
                 => ( ( Y2
                      = ( heap_Time_return_o @ $false ) )
                   => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Leafi @ Uu3 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux3: array_VEBT_VEBTi,Uy3: vEBT_VEBTi] :
                    ( ( X2
                      = ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) )
                   => ( ( Y2
                        = ( heap_Time_return_o @ $true ) )
                     => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ none_P5556105721700978146at_nat @ Uw2 @ Ux3 @ Uy3 ) ) ) )
               => ~ ! [Uz3: product_prod_nat_nat,Va4: nat,Vb2: array_VEBT_VEBTi,Vc2: vEBT_VEBTi] :
                      ( ( X2
                        = ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) )
                     => ( ( Y2
                          = ( heap_Time_return_o @ $false ) )
                       => ~ ( accp_VEBT_VEBTi @ vEBT_V5740978063120863272li_rel @ ( vEBT_Nodei @ ( some_P7363390416028606310at_nat @ Uz3 ) @ Va4 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
thf(fact_9790_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9791_upto_Opelims,axiom,
    ! [X2: int,Xa: int,Y2: list_int] :
      ( ( ( upto @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa )
               => ( Y2
                  = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X2 @ Xa )
               => ( Y2 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_9792_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_9793_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9794_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9795_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9796_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_9797_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_9798_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9799_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9800_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9801_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9802_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_9803_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_9804_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9805_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_9806_upto_Oelims,axiom,
    ! [X2: int,Xa: int,Y2: list_int] :
      ( ( ( upto @ X2 @ Xa )
        = Y2 )
     => ( ( ( ord_less_eq_int @ X2 @ Xa )
         => ( Y2
            = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X2 @ Xa )
         => ( Y2 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9807_upto_Osimps,axiom,
    ( upto
    = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9808_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_9809_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_9810_concat__bit__Suc,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L2 )
      = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).

% concat_bit_Suc
thf(fact_9811_concat__bit__0,axiom,
    ! [K: int,L2: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
      = L2 ) ).

% concat_bit_0
thf(fact_9812_concat__bit__of__zero__2,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_9813_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).

% concat_bit_nonnegative_iff
thf(fact_9814_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L2 ) @ zero_zero_int )
      = ( ord_less_int @ L2 @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_9815_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L2: int] :
      ( ( bit_concat_bit @ N2 @ zero_zero_int @ L2 )
      = ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ).

% concat_bit_of_zero_1
thf(fact_9816_concat__bit__take__bit__eq,axiom,
    ! [N2: nat,B3: int] :
      ( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B3 ) )
      = ( bit_concat_bit @ N2 @ B3 ) ) ).

% concat_bit_take_bit_eq
thf(fact_9817_concat__bit__eq__iff,axiom,
    ! [N2: nat,K: int,L2: int,R: int,S: int] :
      ( ( ( bit_concat_bit @ N2 @ K @ L2 )
        = ( bit_concat_bit @ N2 @ R @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = ( bit_se2923211474154528505it_int @ N2 @ R ) )
        & ( L2 = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_9818_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L2: int,R: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R ) ) ).

% concat_bit_assoc
thf(fact_9819_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).

% concat_bit_eq
thf(fact_9820_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).

% concat_bit_def
thf(fact_9821_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L2: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_9822_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( bit_concat_bit @ N @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_9823_int__set__bit__conv__ops,axiom,
    ( generi8991105624351003935it_int
    = ( ^ [I4: int,N: nat,B2: $o] : ( if_int @ B2 @ ( bit_se1409905431419307370or_int @ I4 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) @ ( bit_se725231765392027082nd_int @ I4 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ) ).

% int_set_bit_conv_ops
thf(fact_9824_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N2: nat] :
      ( ( generi8991105624351003935it_int @ I @ N2 @ $true )
      = ( bit_se1409905431419307370or_int @ I @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ).

% int_set_bit_True_conv_OR
thf(fact_9825_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N2: nat] :
      ( ( generi8991105624351003935it_int @ I @ N2 @ $false )
      = ( bit_se725231765392027082nd_int @ I @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).

% int_set_bit_False_conv_NAND
thf(fact_9826_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_9827_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( ( upt @ zero_zero_nat @ J )
        = nil_nat )
      = ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_9828_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9829_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ J @ K ) )
     => ( ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I @ K ) ) ) ).

% upt_merge
thf(fact_9830_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_9831_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9832_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9833_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_9834_upt__rec,axiom,
    ( upt
    = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9835_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_9836_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B3: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ Ofs )
        @ ( upt @ A3 @ B3 ) )
      = ( upt @ ( plus_plus_nat @ A3 @ Ofs ) @ ( plus_plus_nat @ B3 @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_9837_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9838_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_9839_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_9840_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs2 @ Ys ) )
        = ( ? [K3: nat] :
              ( ( ord_less_eq_nat @ I @ K3 )
              & ( ord_less_eq_nat @ K3 @ J )
              & ( ( upt @ I @ K3 )
                = Xs2 )
              & ( ( upt @ K3 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_9841_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I ) @ ( upt @ I @ J ) )
        = ( upt @ zero_zero_nat @ J ) ) ) ).

% upt_append
thf(fact_9842_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_9843_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_9844_upt__eq__lel__conv,axiom,
    ! [L2: nat,H2: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L2 @ H2 )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L2 @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H2 ) )
        & ( ord_less_eq_nat @ L2 @ I )
        & ( ord_less_nat @ I @ H2 ) ) ) ).

% upt_eq_lel_conv
thf(fact_9845_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X2: nat,Xs2: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X2 @ Xs2 ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X2 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9846_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_9847_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_9848_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_9849_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N2: nat,K: int,L2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2923211474154528505it_int @ N2 @ L2 ) )
     => ( ( map_nat_o @ ( bit_se1146084159140164899it_int @ K ) @ ( upt @ zero_zero_nat @ N2 ) )
        = ( map_nat_o @ ( bit_se1146084159140164899it_int @ L2 ) @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% map_bit_range_eq_if_take_bit_eq
thf(fact_9850_int__sdiv__simps_I2_J,axiom,
    ! [A3: int] :
      ( ( signed6714573509424544716de_int @ A3 @ zero_zero_int )
      = zero_zero_int ) ).

% int_sdiv_simps(2)
thf(fact_9851_sdiv__int__0__div,axiom,
    ! [X2: int] :
      ( ( signed6714573509424544716de_int @ zero_zero_int @ X2 )
      = zero_zero_int ) ).

% sdiv_int_0_div
thf(fact_9852_sdiv__int__div__0,axiom,
    ! [X2: int] :
      ( ( signed6714573509424544716de_int @ X2 @ zero_zero_int )
      = zero_zero_int ) ).

% sdiv_int_div_0
thf(fact_9853_int__sdiv__simps_I1_J,axiom,
    ! [A3: int] :
      ( ( signed6714573509424544716de_int @ A3 @ one_one_int )
      = A3 ) ).

% int_sdiv_simps(1)
thf(fact_9854_int__sdiv__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A3 @ B3 )
          = A3 )
        = ( B3 = one_one_int ) ) ) ).

% int_sdiv_same_is_1
thf(fact_9855_int__sdiv__simps_I3_J,axiom,
    ! [A3: int] :
      ( ( signed6714573509424544716de_int @ A3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A3 ) ) ).

% int_sdiv_simps(3)
thf(fact_9856_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( signed6714573509424544716de_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% sdiv_int_numeral_numeral
thf(fact_9857_int__sdiv__negated__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero_int )
     => ( ( ( signed6714573509424544716de_int @ A3 @ B3 )
          = ( uminus_uminus_int @ A3 ) )
        = ( B3
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% int_sdiv_negated_is_minus1
thf(fact_9858_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( ( signed6714573509424544716de_int @ A3 @ B3 )
       != zero_zero_int )
     => ( ( sgn_sgn_int @ ( signed6714573509424544716de_int @ A3 @ B3 ) )
        = ( sgn_sgn_int @ ( times_times_int @ A3 @ B3 ) ) ) ) ).

% sgn_sdiv_eq_sgn_mult
thf(fact_9859_signed__divide__int__def,axiom,
    ( signed6714573509424544716de_int
    = ( ^ [K3: int,L: int] : ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K3 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% signed_divide_int_def
thf(fact_9860_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U3 )
     => ( ! [I2: nat] :
            ( ( ( ord_less_eq_nat @ U @ I2 )
              & ( ord_less_nat @ I2 @ U3 ) )
           => ~ ( P @ I2 ) )
       => ( ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat2 @ P @ ( upt @ zero_zero_nat @ U3 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_9861_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X: nat] : X
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_9862_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X: int] : X
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_9863_fails__assert_H,axiom,
    ! [P: $o,H2: heap_e7401611519738050253t_unit] :
      ( ( time_f8834461667527620124t_unit @ ( refine_Imp_assert @ P ) @ H2 )
      = ~ P ) ).

% fails_assert'
thf(fact_9864_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9865_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X6: nat > real] :
        ! [J3: nat] :
        ? [M9: nat] :
        ! [M3: nat] :
          ( ( ord_less_eq_nat @ M9 @ M3 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M9 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M3 ) @ ( X6 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9866_VEBT_Osize_I3_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X122 @ 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_9867_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X122: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
      ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X122 @ 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_9868_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_9869_smod__int__range,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero_int )
     => ( member_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ ( uminus_uminus_int @ ( abs_abs_int @ B3 ) ) @ one_one_int ) @ ( minus_minus_int @ ( abs_abs_int @ B3 ) @ one_one_int ) ) ) ) ).

% smod_int_range
thf(fact_9870_smod__int__0__mod,axiom,
    ! [X2: int] :
      ( ( signed6292675348222524329lo_int @ zero_zero_int @ X2 )
      = zero_zero_int ) ).

% smod_int_0_mod
thf(fact_9871_smod__int__mod__0,axiom,
    ! [X2: int] :
      ( ( signed6292675348222524329lo_int @ X2 @ zero_zero_int )
      = X2 ) ).

% smod_int_mod_0
thf(fact_9872_smod__int__numeral__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( signed6292675348222524329lo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ).

% smod_int_numeral_numeral
thf(fact_9873_smod__int__compares_I8_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ B3 @ ( signed6292675348222524329lo_int @ A3 @ B3 ) ) ) ) ).

% smod_int_compares(8)
thf(fact_9874_smod__int__compares_I7_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% smod_int_compares(7)
thf(fact_9875_smod__int__compares_I6_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) ) ) ) ).

% smod_int_compares(6)
thf(fact_9876_smod__int__compares_I4_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).

% smod_int_compares(4)
thf(fact_9877_smod__int__compares_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_eq_int @ zero_zero_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) ) ) ) ).

% smod_int_compares(2)
thf(fact_9878_smod__int__compares_I1_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) @ B3 ) ) ) ).

% smod_int_compares(1)
thf(fact_9879_smod__mod__positive,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
       => ( ( signed6292675348222524329lo_int @ A3 @ B3 )
          = ( modulo_modulo_int @ A3 @ B3 ) ) ) ) ).

% smod_mod_positive
thf(fact_9880_smod__int__compares_I5_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A3 )
     => ( ( ord_less_int @ B3 @ zero_zero_int )
       => ( ord_less_int @ ( signed6292675348222524329lo_int @ A3 @ B3 ) @ ( uminus_uminus_int @ B3 ) ) ) ) ).

% smod_int_compares(5)
thf(fact_9881_smod__int__compares_I3_J,axiom,
    ! [A3: int,B3: int] :
      ( ( ord_less_eq_int @ A3 @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B3 )
       => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( signed6292675348222524329lo_int @ A3 @ B3 ) ) ) ) ).

% smod_int_compares(3)
thf(fact_9882_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T2 @ D )
     => ( vEBT_invar_vebt @ T2 @ D ) ) ).

% valid_eq2
thf(fact_9883_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_9884_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T2 @ D )
     => ( vEBT_VEBT_valid @ T2 @ D ) ) ).

% valid_eq1
thf(fact_9885_uint32_Osize__eq__length,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( type_l796852477590012082l_num1 @ type_N8448461349408098053l_num1 ) ) ).

% uint32.size_eq_length
thf(fact_9886_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv3: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv3 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_9887_len__num0,axiom,
    ( type_l4264026598287037464l_num0
    = ( ^ [Uu4: itself_Numeral_num0] : zero_zero_nat ) ) ).

% len_num0
thf(fact_9888_len__num1,axiom,
    ( type_l4264026598287037465l_num1
    = ( ^ [Uu4: itself_Numeral_num1] : one_one_nat ) ) ).

% len_num1
thf(fact_9889_len__of__finite__1__def,axiom,
    ( type_l31302759751748491nite_1
    = ( ^ [X: itself_finite_1] : one_one_nat ) ) ).

% len_of_finite_1_def
thf(fact_9890_len__of__finite__3__def,axiom,
    ( type_l31302759751748493nite_3
    = ( ^ [X: itself_finite_3] : ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% len_of_finite_3_def
thf(fact_9891_len__of__finite__2__def,axiom,
    ( type_l31302759751748492nite_2
    = ( ^ [X: itself_finite_2] : ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% len_of_finite_2_def
thf(fact_9892_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_9893_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9894_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9895_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_min_nat @ ( suc @ A3 ) @ B3 )
        = ( suc @ A3 ) ) ) ).

% min_Suc_gt(1)
thf(fact_9896_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_min_nat @ B3 @ ( suc @ A3 ) )
        = ( suc @ A3 ) ) ) ).

% min_Suc_gt(2)
thf(fact_9897_min__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).

% min_Suc_numeral
thf(fact_9898_min__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).

% min_numeral_Suc
thf(fact_9899_concat__bit__assoc__sym,axiom,
    ! [M: nat,N2: nat,K: int,L2: int,R: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N2 ) @ L2 @ R ) ) ) ).

% concat_bit_assoc_sym
thf(fact_9900_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_9901_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_9902_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_9903_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N2: nat,K: int,L2: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N2 @ K @ L2 ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ L2 ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_9904_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ M5 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_9905_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M5: nat] : ( suc @ ( ord_min_nat @ N2 @ M5 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_9906_mod__mod__power,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( modulo_modulo_nat @ K @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( ord_min_nat @ M @ N2 ) ) ) ) ).

% mod_mod_power
thf(fact_9907_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_9908_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_9909_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
             => ( ( Deg = Xa )
                & ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( 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 @ TreeList3 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                  @ ( 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 ) )
                        & ! [I4: nat] :
                            ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima2 ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9910_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
             => ~ ( ( Deg = Xa )
                  & ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_VEBT_valid @ X4 @ ( 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 @ TreeList3 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                    @ ( 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 ) )
                          & ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima2 ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9911_int__set__bits__K__False,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu: nat] : $false )
    = zero_zero_int ) ).

% int_set_bits_K_False
thf(fact_9912_int__set__bits__K__True,axiom,
    ( ( bit_bi6516823479961619367ts_int
      @ ^ [Uu: nat] : $true )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% int_set_bits_K_True
thf(fact_9913_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima3: option4927543243414619207at_nat,Deg4: nat,TreeList2: list_VEBT_VEBT,Summary4: vEBT_VEBT,Deg6: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima3 @ Deg4 @ TreeList2 @ Summary4 ) @ Deg6 )
      = ( ( Deg4 = Deg6 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary4 @ ( minus_minus_nat @ Deg4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X6 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
          @ ( 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 ) ) @ Deg4 ) )
                & ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg4 @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg4 ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima3 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9914_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa )
        = Y2 )
     => ( ( ? [Uu3: $o,Uv2: $o] :
              ( X2
              = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
         => ( Y2
            = ( Xa != one_one_nat ) ) )
       => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( X2
                = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
             => ( Y2
                = ( ~ ( ( Deg = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9915_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat,Y2: $o] :
      ( ( ( vEBT_VEBT_valid @ X2 @ Xa )
        = Y2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( Y2
                  = ( Xa = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa ) ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
               => ( ( Y2
                    = ( ( Deg = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9916_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa ) )
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) @ Xa ) )
                 => ~ ( ( Deg = Xa )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( 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 @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                        @ ( 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 ) )
                              & ! [I4: nat] :
                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9917_fi__match__entails,axiom,
    ! [M: list_P8527749157015355191n_assn] :
      ( ! [X3: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ M ) )
         => ( produc7274209992780475162assn_o @ entails @ X3 ) )
     => ( entails @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M ) @ one_one_assn ) @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M ) @ one_one_assn ) ) ) ).

% fi_match_entails
thf(fact_9918_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X2: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X2 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa ) )
       => ( ! [Uu3: $o,Uv2: $o] :
              ( ( X2
                = ( vEBT_Leaf @ Uu3 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu3 @ Uv2 ) @ Xa ) )
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList3: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( X2
                  = ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima2 @ Deg @ TreeList3 @ Summary ) @ Xa ) )
                 => ( ( Deg = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( 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 @ TreeList3 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                      @ ( 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 ) )
                            & ! [I4: nat] :
                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X @ X6 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9919_set__bits__int__def,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F4: nat > $o] :
          ( if_int
          @ ? [N: nat] :
            ! [M3: nat] :
              ( ( ord_less_eq_nat @ N @ M3 )
             => ( ( F4 @ M3 )
                = ( F4 @ N ) ) )
          @ ( bit_ri631733984087533419it_int
            @ ( ord_Least_nat
              @ ^ [N: nat] :
                ! [M3: nat] :
                  ( ( ord_less_eq_nat @ N @ M3 )
                 => ( ( F4 @ M3 )
                    = ( F4 @ N ) ) ) )
            @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
              @ ( map_nat_o @ F4
                @ ( upt @ zero_zero_nat
                  @ ( suc
                    @ ( ord_Least_nat
                      @ ^ [N: nat] :
                        ! [M3: nat] :
                          ( ( ord_less_eq_nat @ N @ M3 )
                         => ( ( F4 @ M3 )
                            = ( F4 @ N ) ) ) ) ) ) ) ) )
          @ zero_zero_int ) ) ) ).

% set_bits_int_def
thf(fact_9920_set__bits__int__unfold_H,axiom,
    ( bit_bi6516823479961619367ts_int
    = ( ^ [F4: nat > $o] :
          ( if_int
          @ ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq_nat @ N @ N12 )
             => ~ ( F4 @ N12 ) )
          @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
            @ ( map_nat_o @ F4
              @ ( upt @ zero_zero_nat
                @ ( ord_Least_nat
                  @ ^ [N: nat] :
                    ! [N12: nat] :
                      ( ( ord_less_eq_nat @ N @ N12 )
                     => ~ ( F4 @ N12 ) ) ) ) ) )
          @ ( if_int
            @ ? [N: nat] :
              ! [N12: nat] :
                ( ( ord_less_eq_nat @ N @ N12 )
               => ( F4 @ N12 ) )
            @ ( bit_ri631733984087533419it_int
              @ ( ord_Least_nat
                @ ^ [N: nat] :
                  ! [N12: nat] :
                    ( ( ord_less_eq_nat @ N @ N12 )
                   => ( F4 @ N12 ) ) )
              @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
                @ ( append_o
                  @ ( map_nat_o @ F4
                    @ ( upt @ zero_zero_nat
                      @ ( ord_Least_nat
                        @ ^ [N: nat] :
                          ! [N12: nat] :
                            ( ( ord_less_eq_nat @ N @ N12 )
                           => ( F4 @ N12 ) ) ) ) )
                  @ ( cons_o @ $true @ nil_o ) ) ) )
            @ zero_zero_int ) ) ) ) ).

% set_bits_int_unfold'
thf(fact_9921_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_9922_Least__Suc,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_9923_Least__Suc2,axiom,
    ! [P: nat > $o,N2: nat,Q: nat > $o,M: nat] :
      ( ( P @ N2 )
     => ( ( Q @ M )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K2: nat] :
                ( ( P @ ( suc @ K2 ) )
                = ( Q @ K2 ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_9924_FI__RESULT__def,axiom,
    ( fI_RESULT
    = ( ^ [M9: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F7: assn] :
          ( ! [X: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ M9 ) )
             => ( produc7274209992780475162assn_o @ entails @ X ) )
         => ( entails @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M9 ) @ one_one_assn ) @ UP ) @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M9 ) @ one_one_assn ) @ UQ ) @ F7 ) ) ) ) ) ).

% FI_RESULT_def
thf(fact_9925_FI__def,axiom,
    ( fi
    = ( ^ [M3: list_P8527749157015355191n_assn,P5: assn,Q4: assn,Up: assn,Uq: assn,F4: assn] :
          ( ! [X: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X @ ( set_Pr1139785259514867910n_assn @ M3 ) )
             => ( produc7274209992780475162assn_o @ entails @ X ) )
         => ( entails @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M3 ) @ one_one_assn ) @ P5 ) @ Up ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M3 ) @ one_one_assn ) @ Q4 ) @ Uq ) @ F4 ) ) ) ) ) ).

% FI_def
thf(fact_9926_bin__last__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ F ) ) )
        = ( F @ zero_zero_nat ) ) ) ).

% bin_last_set_bits
thf(fact_9927_wf__set__bits__int__Suc,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( bit_wf_set_bits_int @ F ) ) ).

% wf_set_bits_int_Suc
thf(fact_9928_wf__set__bits__int__simps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
        ? [N: nat] :
          ( ! [N12: nat] :
              ( ( ord_less_eq_nat @ N @ N12 )
             => ~ ( F4 @ N12 ) )
          | ! [N12: nat] :
              ( ( ord_less_eq_nat @ N @ N12 )
             => ( F4 @ N12 ) ) ) ) ) ).

% wf_set_bits_int_simps
thf(fact_9929_zeros,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N8: nat] :
          ( ( ord_less_eq_nat @ N2 @ N8 )
         => ~ ( F @ N8 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% zeros
thf(fact_9930_wf__set__bits__int_Osimps,axiom,
    ( bit_wf_set_bits_int
    = ( ^ [F4: nat > $o] :
          ( ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq_nat @ N @ N12 )
             => ~ ( F4 @ N12 ) )
          | ? [N: nat] :
            ! [N12: nat] :
              ( ( ord_less_eq_nat @ N @ N12 )
             => ( F4 @ N12 ) ) ) ) ) ).

% wf_set_bits_int.simps
thf(fact_9931_wf__set__bits__int_Ocases,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ! [N4: nat] :
            ~ ! [N7: nat] :
                ( ( ord_less_eq_nat @ N4 @ N7 )
               => ~ ( F @ N7 ) )
       => ~ ! [N4: nat] :
              ~ ! [N7: nat] :
                  ( ( ord_less_eq_nat @ N4 @ N7 )
                 => ( F @ N7 ) ) ) ) ).

% wf_set_bits_int.cases
thf(fact_9932_ones,axiom,
    ! [N2: nat,F: nat > $o] :
      ( ! [N8: nat] :
          ( ( ord_less_eq_nat @ N2 @ N8 )
         => ( F @ N8 ) )
     => ( bit_wf_set_bits_int @ F ) ) ).

% ones
thf(fact_9933_wf__set__bits__int__const,axiom,
    ! [B3: $o] :
      ( bit_wf_set_bits_int
      @ ^ [Uu: nat] : B3 ) ).

% wf_set_bits_int_const
thf(fact_9934_int__set__bits__unfold__BIT,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( bit_bi6516823479961619367ts_int @ F )
        = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( F @ zero_zero_nat ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ) ) ).

% int_set_bits_unfold_BIT
thf(fact_9935_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C5: $o,C6: $o,C7: $o] :
      ( ( ord_less_eq_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C5 @ C6 @ C7 ) )
      = ( ord_less_eq_nat
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C5 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_eq_char_simp
thf(fact_9936_bin__rest__set__bits,axiom,
    ! [F: nat > $o] :
      ( ( bit_wf_set_bits_int @ F )
     => ( ( divide_divide_int @ ( bit_bi6516823479961619367ts_int @ F ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = ( bit_bi6516823479961619367ts_int @ ( comp_nat_o_nat @ F @ suc ) ) ) ) ).

% bin_rest_set_bits
thf(fact_9937_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C5: $o,C6: $o,C7: $o] :
      ( ( ord_less_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) @ ( char2 @ C0 @ C1 @ C22 @ C32 @ C42 @ C5 @ C6 @ C7 ) )
      = ( ord_less_nat
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ B0 @ ( cons_o @ B1 @ ( cons_o @ B22 @ ( cons_o @ B32 @ ( cons_o @ B42 @ ( cons_o @ B52 @ ( cons_o @ B62 @ ( cons_o @ B72 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat )
        @ ( foldr_o_nat
          @ ^ [B2: $o,K3: nat] : ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          @ ( cons_o @ C0 @ ( cons_o @ C1 @ ( cons_o @ C22 @ ( cons_o @ C32 @ ( cons_o @ C42 @ ( cons_o @ C5 @ ( cons_o @ C6 @ ( cons_o @ C7 @ nil_o ) ) ) ) ) ) ) )
          @ zero_zero_nat ) ) ) ).

% less_char_simp
thf(fact_9938_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_9939_less__char__def,axiom,
    ( ord_less_char
    = ( ^ [C12: char,C23: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C12 ) @ ( comm_s629917340098488124ar_nat @ C23 ) ) ) ) ).

% less_char_def
thf(fact_9940_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_9941_less__eq__char__def,axiom,
    ( ord_less_eq_char
    = ( ^ [C12: char,C23: char] : ( ord_less_eq_nat @ ( comm_s629917340098488124ar_nat @ C12 ) @ ( comm_s629917340098488124ar_nat @ C23 ) ) ) ) ).

% less_eq_char_def
thf(fact_9942_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_9943_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X34: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X22 @ X34 @ X43 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9944_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X34: $o,X43: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X1 @ X22 @ X34 @ X43 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_9945_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_9946_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_9947_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_9948_shiftl__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_Sh3965577149348748681tl_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% shiftl_Suc_0
thf(fact_9949_shiftr__Suc__0,axiom,
    ! [N2: nat] :
      ( ( bit_Sh2154871086232339855tr_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).

% shiftr_Suc_0
thf(fact_9950_msb__numeral_I1_J,axiom,
    ! [N2: num] :
      ~ ( most_s5051101344085556sb_int @ ( numeral_numeral_int @ N2 ) ) ).

% msb_numeral(1)
thf(fact_9951_msb__0,axiom,
    ~ ( most_s5051101344085556sb_int @ zero_zero_int ) ).

% msb_0
thf(fact_9952_msb__1,axiom,
    ~ ( most_s5051101344085556sb_int @ one_one_int ) ).

% msb_1
thf(fact_9953_msb__numeral_I2_J,axiom,
    ! [N2: num] : ( most_s5051101344085556sb_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% msb_numeral(2)
thf(fact_9954_msb__bin__rest,axiom,
    ! [X2: int] :
      ( ( most_s5051101344085556sb_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( most_s5051101344085556sb_int @ X2 ) ) ).

% msb_bin_rest
thf(fact_9955_msb__int__def,axiom,
    ( most_s5051101344085556sb_int
    = ( ^ [X: int] : ( ord_less_int @ X @ zero_zero_int ) ) ) ).

% msb_int_def
thf(fact_9956_msb__integer__code,axiom,
    ( most_s5661112943643946085nteger
    = ( ^ [X: code_integer] : ( ord_le6747313008572928689nteger @ X @ zero_z3403309356797280102nteger ) ) ) ).

% msb_integer_code
thf(fact_9957_uint32__msb__test__bit,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X: uint32] : ( bit_se5367290876889521763uint32 @ X @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% uint32_msb_test_bit
thf(fact_9958_msb__uint32__code,axiom,
    ( most_s9063628576841037300uint32
    = ( ^ [X: uint32] : ( uint32_test_bit @ X @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ).

% msb_uint32_code
thf(fact_9959_uint32__test__bit__def,axiom,
    ( uint32_test_bit
    = ( ^ [X: uint32,N: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ X @ N ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
           => ( bit_se5367290876889521763uint32 @ X @ ( code_nat_of_integer @ N ) ) ) ) ) ) ).

% uint32_test_bit_def
thf(fact_9960_test__bit__uint32__code,axiom,
    ( bit_se5367290876889521763uint32
    = ( ^ [X: uint32,N: nat] :
          ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) )
          & ( uint32_test_bit @ X @ ( code_integer_of_nat @ N ) ) ) ) ) ).

% test_bit_uint32_code
thf(fact_9961_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_9962_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_9963_nat__of__integer__numeral,axiom,
    ! [N2: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% nat_of_integer_numeral
thf(fact_9964_nat__of__integer__1,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_1
thf(fact_9965_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_9966_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_9967_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_9968_integer__of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% integer_of_nat_numeral
thf(fact_9969_integer__of__nat__1,axiom,
    ( ( code_integer_of_nat @ one_one_nat )
    = one_one_Code_integer ) ).

% integer_of_nat_1
thf(fact_9970_integer__of__nat__less__0__conv,axiom,
    ! [N2: nat] :
      ~ ( ord_le6747313008572928689nteger @ ( code_integer_of_nat @ N2 ) @ zero_z3403309356797280102nteger ) ).

% integer_of_nat_less_0_conv
thf(fact_9971_nat__of__integer__less__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y2 )
       => ( ( ord_less_nat @ ( code_nat_of_integer @ X2 ) @ ( code_nat_of_integer @ Y2 ) )
          = ( ord_le6747313008572928689nteger @ X2 @ Y2 ) ) ) ) ).

% nat_of_integer_less_iff
thf(fact_9972_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
     => ( ( set_or8404916559141939852nteger @ zero_z3403309356797280102nteger @ U )
        = ( image_1215581382706833972nteger @ semiri4939895301339042750nteger @ ( set_ord_lessThan_nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_integer
thf(fact_9973_integer__set__bit__code,axiom,
    ( bits_integer_set_bit
    = ( ^ [X: code_integer,N: code_integer,B2: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X @ N @ B2 ) @ ( if_Code_integer @ B2 @ ( bit_se1080825931792720795nteger @ X @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N ) @ one_one_Code_integer ) ) @ ( bit_se3949692690581998587nteger @ X @ ( bit_ri7632146776885996613nteger @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N ) @ one_one_Code_integer ) ) ) ) ) ) ) ).

% integer_set_bit_code
thf(fact_9974_uint32__shiftr__def,axiom,
    ( uint32_shiftr
    = ( ^ [X: uint32,N: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
          @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ X @ N )
          @ ( bit_se3964402333458159761uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_shiftr_def
thf(fact_9975_integer__set__bit__def,axiom,
    ( bits_integer_set_bit
    = ( ^ [X: code_integer,N: code_integer,B2: $o] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger ) @ ( undefi1878487536576149250nteger @ X @ N @ B2 ) @ ( generi2397576812484419408nteger @ X @ ( code_nat_of_integer @ N ) @ B2 ) ) ) ) ).

% integer_set_bit_def
thf(fact_9976_shiftr__uint32__code,axiom,
    ( bit_se3964402333458159761uint32
    = ( ^ [N: nat,X: uint32] : ( if_uint32 @ ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftr @ X @ ( code_integer_of_nat @ N ) ) @ zero_zero_uint32 ) ) ) ).

% shiftr_uint32_code
thf(fact_9977_uint32__shiftl__def,axiom,
    ( uint32_shiftl
    = ( ^ [X: uint32,N: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
          @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ X @ N )
          @ ( bit_se5742574853984576102uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_shiftl_def
thf(fact_9978_uint32__set__bit__def,axiom,
    ( uint32_set_bit
    = ( ^ [X: uint32,N: code_integer,B2: $o] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
          @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ X @ N @ B2 )
          @ ( generi1993664874377053279uint32 @ X @ ( code_nat_of_integer @ N ) @ B2 ) ) ) ) ).

% uint32_set_bit_def
thf(fact_9979_set__bit__uint32__code,axiom,
    ( generi1993664874377053279uint32
    = ( ^ [X: uint32,N: nat,B2: $o] : ( if_uint32 @ ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_set_bit @ X @ ( code_integer_of_nat @ N ) @ B2 ) @ X ) ) ) ).

% set_bit_uint32_code
thf(fact_9980_shiftl__uint32__code,axiom,
    ( bit_se5742574853984576102uint32
    = ( ^ [N: nat,X: uint32] : ( if_uint32 @ ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_shiftl @ X @ ( code_integer_of_nat @ N ) ) @ zero_zero_uint32 ) ) ) ).

% shiftl_uint32_code
thf(fact_9981_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9982_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9983_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9984_zero__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% zero_integer.rep_eq
thf(fact_9985_int__of__integer__numeral,axiom,
    ! [K: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% int_of_integer_numeral
thf(fact_9986_one__integer_Orep__eq,axiom,
    ( ( code_int_of_integer @ one_one_Code_integer )
    = one_one_int ) ).

% one_integer.rep_eq
thf(fact_9987_divide__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X2 @ Xa ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% divide_integer.rep_eq
thf(fact_9988_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9989_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).

% integer_less_iff
thf(fact_9990_int__of__integer__less__iff,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Y2 ) )
      = ( ord_le6747313008572928689nteger @ X2 @ Y2 ) ) ).

% int_of_integer_less_iff
thf(fact_9991_int__of__integer__pow,axiom,
    ! [X2: code_integer,N2: nat] :
      ( ( code_int_of_integer @ ( power_8256067586552552935nteger @ X2 @ N2 ) )
      = ( power_power_int @ ( code_int_of_integer @ X2 ) @ N2 ) ) ).

% int_of_integer_pow
thf(fact_9992_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9993_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9994_bin__last__integer_Orep__eq,axiom,
    ( bits_b8758750999018896077nteger
    = ( ^ [X: code_integer] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X ) ) ) ) ).

% bin_last_integer.rep_eq
thf(fact_9995_bin__rest__integer_Orep__eq,axiom,
    ! [X2: code_integer] :
      ( ( code_int_of_integer @ ( bits_b2549910563261871055nteger @ X2 ) )
      = ( divide_divide_int @ ( code_int_of_integer @ X2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% bin_rest_integer.rep_eq
thf(fact_9996_Bit__integer_Orep__eq,axiom,
    ! [X2: code_integer,Xa: $o] :
      ( ( code_int_of_integer @ ( bits_Bit_integer @ X2 @ Xa ) )
      = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ Xa ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ X2 ) ) ) ) ).

% Bit_integer.rep_eq
thf(fact_9997_uint32__shiftr__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N2 ) )
          = ( rep_uint322 @ ( undefi8952517107220742160uint32 @ bit_se3964402333458159761uint32 @ W @ N2 ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftr @ W @ N2 ) )
          = ( bit_se5176125413884933531l_num1 @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftr_code
thf(fact_9998_uint32__shiftl__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N2 ) )
          = ( rep_uint322 @ ( undefi8952517107220742160uint32 @ bit_se5742574853984576102uint32 @ W @ N2 ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_shiftl @ W @ N2 ) )
          = ( bit_se837345729053750000l_num1 @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_shiftl_code
thf(fact_9999_zero__uint32_Orep__eq,axiom,
    ( ( rep_uint322 @ zero_zero_uint32 )
    = zero_z3563351764282998399l_num1 ) ).

% zero_uint32.rep_eq
thf(fact_10000_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( ( dvd_dvd_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ one ) ) @ P2 )
      = ( dvd_dv6812691276156420380l_num1 @ ( numera7442385471795722001l_num1 @ ( bit0 @ one ) ) @ ( rep_uint322 @ P2 ) ) ) ).

% uint32.even_iff_word_of
thf(fact_10001_size__uint32_Orep__eq,axiom,
    ( size_size_uint32
    = ( ^ [X: uint32] : ( size_s8261804613246490634l_num1 @ ( rep_uint322 @ X ) ) ) ) ).

% size_uint32.rep_eq
thf(fact_10002_uint32_Osize__eq__word__of,axiom,
    ( size_size_uint32
    = ( ^ [P5: uint32] : ( size_s8261804613246490634l_num1 @ ( rep_uint322 @ P5 ) ) ) ) ).

% uint32.size_eq_word_of
thf(fact_10003_uint32_Oless__iff__word__of,axiom,
    ( ord_less_uint32
    = ( ^ [P5: uint32,Q4: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint322 @ P5 ) @ ( rep_uint322 @ Q4 ) ) ) ) ).

% uint32.less_iff_word_of
thf(fact_10004_less__uint32_Orep__eq,axiom,
    ( ord_less_uint32
    = ( ^ [X: uint32,Xa4: uint32] : ( ord_le750835935415966154l_num1 @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Xa4 ) ) ) ) ).

% less_uint32.rep_eq
thf(fact_10005_uint32__test__bit__code,axiom,
    ( uint32_test_bit
    = ( ^ [W2: uint32,N: code_integer] :
          ( ( ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
              | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
           => ( undefi6981832269580975664eger_o @ bit_se5367290876889521763uint32 @ W2 @ N ) )
          & ( ~ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
                | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N ) )
           => ( bit_se6859397288646540909l_num1 @ ( rep_uint322 @ W2 ) @ ( code_nat_of_integer @ N ) ) ) ) ) ) ).

% uint32_test_bit_code
thf(fact_10006_uint32__set__bit__code,axiom,
    ! [N2: code_integer,W: uint32,B3: $o] :
      ( ( ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
          | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B3 ) )
          = ( rep_uint322 @ ( undefi8537048349889504752uint32 @ generi1993664874377053279uint32 @ W @ N2 @ B3 ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
            | ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_set_bit @ W @ N2 @ B3 ) )
          = ( generi5268133209446125161l_num1 @ ( rep_uint322 @ W ) @ ( code_nat_of_integer @ N2 ) @ B3 ) ) ) ) ).

% uint32_set_bit_code
thf(fact_10007_Uint32__signed__code,axiom,
    ! [I: code_integer] :
      ( ( ( ( ord_le6747313008572928689nteger @ I @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
       => ( ( rep_uint322 @ ( uint32_signed @ I ) )
          = ( rep_uint322 @ ( undefi2040150642751712519uint32 @ uint322 @ I ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ I @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ I ) )
       => ( ( rep_uint322 @ ( uint32_signed @ I ) )
          = ( ring_17408606157368542149l_num1 @ ( code_I935103866777955880mbolic @ I ) ) ) ) ) ).

% Uint32_signed_code
thf(fact_10008_uint32_Oset__bits__aux__code,axiom,
    ( set_bits_aux_uint32
    = ( ^ [F4: nat > $o,N: nat,W2: uint32] : ( if_uint32 @ ( N = zero_zero_nat ) @ W2 @ ( set_bits_aux_uint32 @ F4 @ ( minus_minus_nat @ N @ one_one_nat ) @ ( bit_se2966626333419230250uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ W2 ) @ ( if_uint32 @ ( F4 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ one_one_uint32 @ zero_zero_uint32 ) ) ) ) ) ) ).

% uint32.set_bits_aux_code
thf(fact_10009_zero__uint32_Orsp,axiom,
    zero_z3563351764282998399l_num1 = zero_z3563351764282998399l_num1 ).

% zero_uint32.rsp
thf(fact_10010_uint32_Oset__bits__code,axiom,
    ( bit_bi705532357378895591uint32
    = ( ^ [P4: nat > $o] : ( set_bits_aux_uint32 @ P4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ zero_zero_uint32 ) ) ) ).

% uint32.set_bits_code
thf(fact_10011_int__of__integer__symbolic__aux__code_I1_J,axiom,
    ( ( code_I935103866777955880mbolic @ zero_z3403309356797280102nteger )
    = zero_zero_int ) ).

% int_of_integer_symbolic_aux_code(1)
thf(fact_10012_uint32__divmod__code,axiom,
    ( uint32_divmod
    = ( ^ [X: uint32,Y: uint32] : ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ ( numera9087168376688890119uint32 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ Y ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_uint32 @ X @ Y ) @ ( produc1400373151660368625uint32 @ zero_zero_uint32 @ X ) @ ( produc1400373151660368625uint32 @ one_one_uint32 @ ( minus_minus_uint32 @ X @ Y ) ) ) @ ( if_Pro1135515155860407935uint32 @ ( Y = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( div0_uint32 @ X ) @ ( mod0_uint32 @ X ) ) @ ( if_Pro1135515155860407935uint32 @ ( ord_less_eq_uint32 @ Y @ ( minus_minus_uint32 @ X @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X ) @ Y ) ) @ Y ) ) ) @ ( produc1400373151660368625uint32 @ ( plus_plus_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X ) @ Y ) ) @ one_one_uint32 ) @ ( minus_minus_uint32 @ ( minus_minus_uint32 @ X @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X ) @ Y ) ) @ Y ) ) @ Y ) ) @ ( produc1400373151660368625uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X ) @ Y ) ) @ ( minus_minus_uint32 @ X @ ( times_times_uint32 @ ( bit_se5742574853984576102uint32 @ one_one_nat @ ( uint32_sdiv @ ( bit_se3964402333458159761uint32 @ one_one_nat @ X ) @ Y ) ) @ Y ) ) ) ) ) ) ) ) ).

% uint32_divmod_code
thf(fact_10013_uint32__divmod__def,axiom,
    ( uint32_divmod
    = ( ^ [X: uint32,Y: uint32] : ( if_Pro1135515155860407935uint32 @ ( Y = zero_zero_uint32 ) @ ( produc1400373151660368625uint32 @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X @ zero_zero_uint32 ) @ ( undefi332904144742839227uint32 @ modulo_modulo_uint32 @ X @ zero_zero_uint32 ) ) @ ( produc1400373151660368625uint32 @ ( divide_divide_uint32 @ X @ Y ) @ ( modulo_modulo_uint32 @ X @ Y ) ) ) ) ) ).

% uint32_divmod_def
thf(fact_10014_div0__uint32__def,axiom,
    ( div0_uint32
    = ( ^ [X: uint32] : ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X @ zero_zero_uint32 ) ) ) ).

% div0_uint32_def
thf(fact_10015_mod0__uint32__def,axiom,
    ( mod0_uint32
    = ( ^ [X: uint32] : ( undefi332904144742839227uint32 @ modulo_modulo_uint32 @ X @ zero_zero_uint32 ) ) ) ).

% mod0_uint32_def
thf(fact_10016_uint32__sdiv__code,axiom,
    ! [Y2: uint32,X2: uint32] :
      ( ( ( Y2 = zero_zero_uint32 )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X2 @ Y2 ) )
          = ( rep_uint322 @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X2 @ zero_zero_uint32 ) ) ) )
      & ( ( Y2 != zero_zero_uint32 )
       => ( ( rep_uint322 @ ( uint32_sdiv @ X2 @ Y2 ) )
          = ( signed6753297604338940182l_num1 @ ( rep_uint322 @ X2 ) @ ( rep_uint322 @ Y2 ) ) ) ) ) ).

% uint32_sdiv_code
thf(fact_10017_sshiftr__uint32__code,axiom,
    ( signed489701013188660438uint32
    = ( ^ [N: nat,X: uint32] : ( if_uint32 @ ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) @ ( uint32_sshiftr @ X @ ( code_integer_of_nat @ N ) ) @ ( if_uint32 @ ( bit_se5367290876889521763uint32 @ X @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( uminus_uminus_uint32 @ one_one_uint32 ) @ zero_zero_uint32 ) ) ) ) ).

% sshiftr_uint32_code
thf(fact_10018_integer__shiftl__def,axiom,
    ( bits_integer_shiftl
    = ( ^ [X: code_integer,N: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X @ N ) @ ( bit_se7788150548672797655nteger @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% integer_shiftl_def
thf(fact_10019_integer__shiftl__code_I2_J,axiom,
    ! [X2: code_integer] :
      ( ( bits_integer_shiftl @ X2 @ zero_z3403309356797280102nteger )
      = X2 ) ).

% integer_shiftl_code(2)
thf(fact_10020_uint32__sshiftr__def,axiom,
    ( uint32_sshiftr
    = ( ^ [X: uint32,N: code_integer] :
          ( if_uint32
          @ ( ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N ) )
          @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N @ X )
          @ ( signed489701013188660438uint32 @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% uint32_sshiftr_def
thf(fact_10021_uint32__sshiftr__code,axiom,
    ! [N2: code_integer,W: uint32] :
      ( ( ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
          | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N2 ) )
          = ( rep_uint322 @ ( undefi7330133036835070352uint32 @ signed489701013188660438uint32 @ N2 @ W ) ) ) )
      & ( ~ ( ( ord_le6747313008572928689nteger @ N2 @ zero_z3403309356797280102nteger )
            | ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) @ N2 ) )
       => ( ( rep_uint322 @ ( uint32_sshiftr @ W @ N2 ) )
          = ( signed5000768011106662067l_num1 @ ( code_nat_of_integer @ N2 ) @ ( rep_uint322 @ W ) ) ) ) ) ).

% uint32_sshiftr_code
thf(fact_10022_div__uint32__code,axiom,
    ( divide_divide_uint32
    = ( ^ [X: uint32,Y: uint32] : ( if_uint32 @ ( Y = zero_zero_uint32 ) @ zero_zero_uint32 @ ( uint32_div @ X @ Y ) ) ) ) ).

% div_uint32_code
thf(fact_10023_mod__uint32__code,axiom,
    ( modulo_modulo_uint32
    = ( ^ [X: uint32,Y: uint32] : ( if_uint32 @ ( Y = zero_zero_uint32 ) @ X @ ( uint32_mod @ X @ Y ) ) ) ) ).

% mod_uint32_code
thf(fact_10024_integer__shiftr__def,axiom,
    ( bits_integer_shiftr
    = ( ^ [X: code_integer,N: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ N @ zero_z3403309356797280102nteger ) @ ( undefi8133104259855420269nteger @ X @ N ) @ ( bit_se3928097537394005634nteger @ ( code_nat_of_integer @ N ) @ X ) ) ) ) ).

% integer_shiftr_def
thf(fact_10025_integer__shiftr__code_I2_J,axiom,
    ! [X2: code_integer] :
      ( ( bits_integer_shiftr @ X2 @ zero_z3403309356797280102nteger )
      = X2 ) ).

% integer_shiftr_code(2)
thf(fact_10026_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L ) )
                @ ( code_divmod_abs @ K3 @ L )
                @ ( produc6916734918728496179nteger
                  @ ^ [R6: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R6 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S2 ) ) )
                  @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_10027_integer__of__uint32__code,axiom,
    ( integer_of_uint32
    = ( ^ [N: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( bit_se1080825931792720795nteger @ ( intege5370686899274169573signed @ ( bit_se6294004230839889034uint32 @ N @ ( numera9087168376688890119uint32 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( intege5370686899274169573signed @ N ) ) ) ) ).

% integer_of_uint32_code
thf(fact_10028_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_10029_integer__of__uint32__signed__def,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( undefi3580195557576403463nteger @ integer_of_uint32 @ N ) @ ( integer_of_uint32 @ N ) ) ) ) ).

% integer_of_uint32_signed_def
thf(fact_10030_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_10031_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
              @ ( produc6916734918728496179nteger
                @ ^ [R6: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R6 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S2 ) ) )
                @ ( code_divmod_abs @ K3 @ L ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R6: code_integer,S2: code_integer] : ( if_Pro6119634080678213985nteger @ ( S2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R6 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S2 ) ) )
                    @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_10032_integer__of__uint32__signed__code,axiom,
    ( intege5370686899274169573signed
    = ( ^ [N: uint32] : ( if_Code_integer @ ( bit_se5367290876889521763uint32 @ N @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( undefi3580195557576403463nteger @ integer_of_uint32 @ N ) @ ( code_integer_of_int @ ( semiri7338730514057886004m1_int @ ( rep_uint32 @ N ) ) ) ) ) ) ).

% integer_of_uint32_signed_code
thf(fact_10033_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R6: code_integer,S2: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R6 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R6 ) @ S2 ) ) @ ( S2 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_10034_Rep__uint32_H__code,axiom,
    ( rep_uint32
    = ( ^ [X: uint32] : ( bit_bi5746210779246519537l_num1 @ ( bit_se5367290876889521763uint32 @ X ) ) ) ) ).

% Rep_uint32'_code
thf(fact_10035_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_10036_char__of__integer__code,axiom,
    ( char_of_integer
    = ( ^ [K3: code_integer] :
          ( produc4188289175737317920o_char
          @ ^ [Q0: code_integer,B02: $o] :
              ( produc4188289175737317920o_char
              @ ^ [Q1: code_integer,B12: $o] :
                  ( produc4188289175737317920o_char
                  @ ^ [Q22: code_integer,B23: $o] :
                      ( produc4188289175737317920o_char
                      @ ^ [Q32: code_integer,B33: $o] :
                          ( produc4188289175737317920o_char
                          @ ^ [Q42: code_integer,B43: $o] :
                              ( produc4188289175737317920o_char
                              @ ^ [Q52: code_integer,B53: $o] :
                                  ( produc4188289175737317920o_char
                                  @ ^ [Q62: code_integer,B63: $o] :
                                      ( produc4188289175737317920o_char
                                      @ ^ [Uu: code_integer] : ( char2 @ B02 @ B12 @ B23 @ B33 @ B43 @ B53 @ B63 )
                                      @ ( code_bit_cut_integer @ Q62 ) )
                                  @ ( code_bit_cut_integer @ Q52 ) )
                              @ ( code_bit_cut_integer @ Q42 ) )
                          @ ( code_bit_cut_integer @ Q32 ) )
                      @ ( code_bit_cut_integer @ Q22 ) )
                  @ ( code_bit_cut_integer @ Q1 ) )
              @ ( code_bit_cut_integer @ Q0 ) )
          @ ( code_bit_cut_integer @ K3 ) ) ) ) ).

% char_of_integer_code
thf(fact_10037_uint32__sdiv__def,axiom,
    ( uint32_sdiv
    = ( ^ [X: uint32,Y: uint32] : ( if_uint32 @ ( Y = zero_zero_uint32 ) @ ( undefi332904144742839227uint32 @ divide_divide_uint32 @ X @ zero_zero_uint32 ) @ ( abs_uint32 @ ( signed6753297604338940182l_num1 @ ( rep_uint322 @ X ) @ ( rep_uint322 @ Y ) ) ) ) ) ) ).

% uint32_sdiv_def
thf(fact_10038_less__uint32_Oabs__eq,axiom,
    ! [Xa: word_N3645301735248828278l_num1,X2: word_N3645301735248828278l_num1] :
      ( ( ord_less_uint32 @ ( abs_uint32 @ Xa ) @ ( abs_uint32 @ X2 ) )
      = ( ord_le750835935415966154l_num1 @ Xa @ X2 ) ) ).

% less_uint32.abs_eq
thf(fact_10039_size__uint32_Oabs__eq,axiom,
    ! [X2: word_N3645301735248828278l_num1] :
      ( ( size_size_uint32 @ ( abs_uint32 @ X2 ) )
      = ( size_s8261804613246490634l_num1 @ X2 ) ) ).

% size_uint32.abs_eq
thf(fact_10040_zero__uint32__def,axiom,
    ( zero_zero_uint32
    = ( abs_uint32 @ zero_z3563351764282998399l_num1 ) ) ).

% zero_uint32_def
thf(fact_10041_DERIV__real__root__generic,axiom,
    ! [N2: nat,X2: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X2 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X2 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X2 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_10042_DERIV__even__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X2 @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_10043_inj__on__diff__nat,axiom,
    ! [N3: set_nat,K: nat] :
      ( ! [N4: nat] :
          ( ( member_nat @ N4 @ N3 )
         => ( ord_less_eq_nat @ K @ N4 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N3 ) ) ).

% inj_on_diff_nat
thf(fact_10044_inj__Suc,axiom,
    ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).

% inj_Suc
thf(fact_10045_DERIV__mirror,axiom,
    ! [F: real > real,Y2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ Y2 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X2 ) @ top_top_set_real ) )
      = ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( F @ ( uminus_uminus_real @ X ) )
        @ ( uminus_uminus_real @ Y2 )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_mirror
thf(fact_10046_DERIV__ln__divide,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_10047_DERIV__const__average,axiom,
    ! [A3: real,B3: real,V: real > real,K: real] :
      ( ( A3 != B3 )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A3 @ B3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A3 ) @ ( V @ B3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_10048_DERIV__ln,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_10049_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D2 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_10050_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D2 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_10051_DERIV__isconst__all,axiom,
    ! [F: real > real,X2: real,Y2: real] :
      ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( F @ X2 )
        = ( F @ Y2 ) ) ) ).

% DERIV_isconst_all
thf(fact_10052_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L2: real,X2: real,S4: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S4 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S4 )
                 => ( ( ord_less_real @ H4 @ D2 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_10053_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L2: real,X2: real,S4: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S4 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S4 )
                 => ( ( ord_less_real @ H4 @ D2 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_10054_DERIV__pow,axiom,
    ! [N2: nat,X2: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X2 @ S ) ) ).

% DERIV_pow
thf(fact_10055_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X2: real,S4: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S4 ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S4 )
                 => ( ( ord_less_real @ H4 @ D2 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_10056_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X2: real,S4: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ S4 ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S4 )
                 => ( ( ord_less_real @ H4 @ D2 )
                   => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_10057_deriv__nonneg__imp__mono,axiom,
    ! [A3: real,B3: real,G: real > real,G2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A3 @ B3 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
       => ( ( ord_less_eq_real @ A3 @ B3 )
         => ( ord_less_eq_real @ ( G @ A3 ) @ ( G @ B3 ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_10058_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A3: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A3 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B3 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_10059_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A3: real,B3: real,F: real > real] :
      ( ( ord_less_eq_real @ A3 @ B3 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A3 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B3 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B3 ) @ ( F @ A3 ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_10060_DERIV__local__const,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ D )
             => ( ( F @ X2 )
                = ( F @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_10061_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D2 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_10062_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L2: real,X2: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [D2: real] :
            ( ( ord_less_real @ zero_zero_real @ D2 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D2 )
                 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_10063_DERIV__local__min,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_10064_DERIV__local__max,axiom,
    ! [F: real > real,L2: real,X2: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X2 ) ) )
         => ( L2 = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_10065_DERIV__pos__imp__increasing,axiom,
    ! [A3: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A3 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B3 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A3 ) @ ( F @ B3 ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_10066_DERIV__neg__imp__decreasing,axiom,
    ! [A3: real,B3: real,F: real > real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A3 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B3 )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B3 ) @ ( F @ A3 ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_10067_MVT2,axiom,
    ! [A3: real,B3: real,F: real > real,F3: real > real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A3 @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B3 )
             => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z4: real] :
            ( ( ord_less_real @ A3 @ Z4 )
            & ( ord_less_real @ Z4 @ B3 )
            & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) )
              = ( times_times_real @ ( minus_minus_real @ B3 @ A3 ) @ ( F3 @ Z4 ) ) ) ) ) ) ).

% MVT2
thf(fact_10068_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X2: real,N2: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N2 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X2 ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_10069_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_10070_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_10071_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X2: real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R )
          @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X2 ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_10072_DERIV__log,axiom,
    ! [X2: real,B3: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ ( log @ B3 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B3 ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_10073_DERIV__powr,axiom,
    ! [G: real > real,M: real,X2: real,F: real > real,R: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X2 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G @ X2 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_10074_DERIV__real__sqrt,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ zero_zero_real @ X2 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_10075_DERIV__arctan,axiom,
    ! [X2: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_10076_arsinh__real__has__field__derivative,axiom,
    ! [X2: real,A4: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A4 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_10077_DERIV__real__sqrt__generic,axiom,
    ! [X2: real,D4: real] :
      ( ( X2 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X2 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X2 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_10078_arcosh__real__has__field__derivative,axiom,
    ! [X2: real,A4: set_real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A4 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_10079_artanh__real__has__field__derivative,axiom,
    ! [X2: real,A4: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A4 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_10080_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_10081_DERIV__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X2 )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_10082_DERIV__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_10083_DERIV__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_10084_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10085_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
         => ( ! [X3: nat] :
                ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X3 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_10086_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
          & ( ( F @ X2 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_10087_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X2: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_10088_DERIV__odd__real__root,axiom,
    ! [N2: nat,X2: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X2 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X2 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_10089_Maclaurin__minus,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H2 @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T3: real] :
                ( ( ( ord_less_nat @ M4 @ N2 )
                  & ( ord_less_eq_real @ H2 @ T3 )
                  & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ H2 @ T3 )
                & ( ord_less_real @ T3 @ zero_zero_real )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [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 @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_10090_Maclaurin2,axiom,
    ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T3 )
                & ( ord_less_eq_real @ T3 @ H2 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ? [T3: real] :
              ( ( ord_less_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ H2 )
              & ( ( F @ H2 )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [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 @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_10091_Maclaurin,axiom,
    ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M4: nat,T3: real] :
                ( ( ( ord_less_nat @ M4 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
                  & ( ord_less_eq_real @ T3 @ H2 ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ T3 )
                & ( ord_less_real @ T3 @ H2 )
                & ( ( F @ H2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [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 @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_10092_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X2 != zero_zero_real )
         => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
                & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
                & ( ( F @ X2 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_10093_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X2: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M4: nat,T3: real] :
            ( ( ( ord_less_nat @ M4 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X2 ) )
            & ( ( F @ X2 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X2 @ M3 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_10094_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real,X2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N2 )
                & ( ord_less_eq_real @ A3 @ T3 )
                & ( ord_less_eq_real @ T3 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A3 @ C )
           => ( ( ord_less_eq_real @ C @ B3 )
             => ( ( ord_less_eq_real @ A3 @ X2 )
               => ( ( ord_less_eq_real @ X2 @ B3 )
                 => ( ( X2 != C )
                   => ? [T3: real] :
                        ( ( ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ X2 @ T3 )
                            & ( ord_less_real @ T3 @ C ) ) )
                        & ( ~ ( ord_less_real @ X2 @ C )
                         => ( ( ord_less_real @ C @ T3 )
                            & ( ord_less_real @ T3 @ X2 ) ) )
                        & ( ( F @ X2 )
                          = ( 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 @ X2 @ C ) @ M3 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_10095_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N2 )
                & ( ord_less_eq_real @ A3 @ T3 )
                & ( ord_less_eq_real @ T3 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A3 @ C )
           => ( ( ord_less_real @ C @ B3 )
             => ? [T3: real] :
                  ( ( ord_less_real @ C @ T3 )
                  & ( ord_less_real @ T3 @ B3 )
                  & ( ( F @ B3 )
                    = ( 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 @ B3 @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_10096_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A3: real,B3: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M4: nat,T3: real] :
              ( ( ( ord_less_nat @ M4 @ N2 )
                & ( ord_less_eq_real @ A3 @ T3 )
                & ( ord_less_eq_real @ T3 @ B3 ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A3 @ C )
           => ( ( ord_less_eq_real @ C @ B3 )
             => ? [T3: real] :
                  ( ( ord_less_real @ A3 @ T3 )
                  & ( ord_less_real @ T3 @ C )
                  & ( ( F @ A3 )
                    = ( 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 @ A3 @ C ) @ M3 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A3 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_10097_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10098_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B6: real] :
      ( ! [M4: nat,T3: real] :
          ( ( ( ord_less_nat @ M4 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ H2 ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M2: nat,T8: real] :
            ( ( ( ord_less_nat @ M2 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T8 )
              & ( ord_less_eq_real @ T8 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M2 ) ) )
                    @ ( times_times_real @ B6 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T8 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T8 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B6 @ ( divide_divide_real @ ( power_power_real @ T8 @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T8 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_10099_DERIV__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X10: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X10 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_10100_DERIV__power__series_H,axiom,
    ! [R4: real,F: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X3 @ N ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
       => ( ( ord_less_real @ zero_zero_real @ R4 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ ( suc @ N ) ) ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_10101_DERIV__isconst3,axiom,
    ! [A3: real,B3: real,X2: real,Y2: real,F: real > real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
       => ( ( member_real @ Y2 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
           => ( ( F @ X2 )
              = ( F @ Y2 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_10102_DERIV__series_H,axiom,
    ! [F: real > nat > real,F3: real > nat > real,X0: real,A3: real,B3: real,L5: nat > real] :
      ( ! [N4: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N4 )
          @ ( F3 @ X0 @ N4 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
           => ( summable_real @ ( F @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
         => ( ( summable_real @ ( F3 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N4: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A3 @ B3 ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y3 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F3 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_10103_LIM__fun__less__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L2 @ zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_10104_LIM__fun__not__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L2 != zero_zero_real )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ( F @ X4 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_10105_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L2: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L2 )
       => ? [R3: real] :
            ( ( ord_less_real @ zero_zero_real @ R3 )
            & ! [X4: real] :
                ( ( ( X4 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_10106_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_10107_isCont__inverse__function2,axiom,
    ! [A3: real,X2: real,B3: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A3 @ X2 )
     => ( ( ord_less_real @ X2 @ B3 )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A3 @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B3 )
               => ( ( G @ ( F @ Z4 ) )
                  = Z4 ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_eq_real @ A3 @ Z4 )
               => ( ( ord_less_eq_real @ Z4 @ B3 )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_10108_isCont__ln,axiom,
    ! [X2: real] :
      ( ( X2 != zero_zero_real )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ ln_ln_real ) ) ).

% isCont_ln
thf(fact_10109_isCont__arcosh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_10110_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_10111_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X2: real,A3: real,B3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X2 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A3 @ X2 )
         => ( ( ord_less_real @ X2 @ B3 )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A3 @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B3 )
                   => ( ( F @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10112_isCont__arccos,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_10113_isCont__arcsin,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_10114_LIM__less__bound,axiom,
    ! [B3: real,X2: real,F: real > real] :
      ( ( ord_less_real @ B3 @ X2 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B3 @ X2 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10115_isCont__artanh,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_10116_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_10117_isCont__inverse__function,axiom,
    ! [D: real,X2: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
           => ( ( G @ ( F @ Z4 ) )
              = Z4 ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10118_GMVT_H,axiom,
    ! [A3: real,B3: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
      ( ( ord_less_real @ A3 @ B3 )
     => ( ! [Z4: real] :
            ( ( ord_less_eq_real @ A3 @ Z4 )
           => ( ( ord_less_eq_real @ Z4 @ B3 )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z4: real] :
              ( ( ord_less_eq_real @ A3 @ Z4 )
             => ( ( ord_less_eq_real @ Z4 @ B3 )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z4: real] :
                ( ( ord_less_real @ A3 @ Z4 )
               => ( ( ord_less_real @ Z4 @ B3 )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
           => ( ! [Z4: real] :
                  ( ( ord_less_real @ A3 @ Z4 )
                 => ( ( ord_less_real @ Z4 @ B3 )
                   => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A3 @ C2 )
                  & ( ord_less_real @ C2 @ B3 )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A3 ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B3 ) @ ( G @ A3 ) ) @ ( F3 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10119_summable__Leibniz_I3_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A3 )
       => ( ( ord_less_real @ ( A3 @ zero_zero_nat ) @ zero_zero_real )
         => ! [N5: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10120_summable__Leibniz_I2_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A3 )
       => ( ( ord_less_real @ zero_zero_real @ ( A3 @ zero_zero_nat ) )
         => ! [N5: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10121_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_10122_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_10123_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( root @ N @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_10124_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
         => ( ( filterlim_nat_real
              @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L3: real] :
                ( ! [N5: nat] : ( ord_less_eq_real @ ( F @ N5 ) @ L3 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat )
                & ! [N5: nat] : ( ord_less_eq_real @ L3 @ ( G @ N5 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L3 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_10125_LIMSEQ__inverse__zero,axiom,
    ! [X9: nat > real] :
      ( ! [R3: real] :
        ? [N9: nat] :
        ! [N4: nat] :
          ( ( ord_less_eq_nat @ N9 @ N4 )
         => ( ord_less_real @ R3 @ ( X9 @ N4 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X9 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10126_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10127_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_10128_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( root @ N @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_10129_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10130_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L2: real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
       => ( ! [E: real] :
              ( ( ord_less_real @ zero_zero_real @ E )
             => ? [N5: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_10131_LIMSEQ__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X2 )
     => ( ( ord_less_real @ X2 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10132_LIMSEQ__divide__realpow__zero,axiom,
    ! [X2: real,A3: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( divide_divide_real @ A3 @ ( power_power_real @ X2 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10133_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_10134_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_10135_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X2: real] :
      ( ( ord_less_real @ one_one_real @ X2 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10136_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10137_tendsto__exp__limit__sequentially,axiom,
    ! [X2: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10138_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10139_summable__Leibniz_I1_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A3 )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A3 @ N ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10140_summable,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A3 @ N ) ) ) ) ) ) ).

% summable
thf(fact_10141_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10142_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10143_summable__Leibniz_I4_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A3 )
       => ( filterlim_nat_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10144_zeroseq__arctan__series,axiom,
    ! [X2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_10145_summable__Leibniz_H_I2_J,axiom,
    ! [A3: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10146_summable__Leibniz_H_I3_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10147_sums__alternating__upper__lower,axiom,
    ! [A3: nat > real] :
      ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L3: real] :
              ( ! [N5: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
                  @ L3 )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat )
              & ! [N5: nat] :
                  ( ord_less_eq_real @ L3
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L3 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10148_summable__Leibniz_I5_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A3 )
       => ( filterlim_nat_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10149_summable__Leibniz_H_I5_J,axiom,
    ! [A3: nat > real] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10150_summable__Leibniz_H_I4_J,axiom,
    ! [A3: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A3 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A3 @ N4 ) )
       => ( ! [N4: nat] : ( ord_less_eq_real @ ( A3 @ ( suc @ N4 ) ) @ ( A3 @ N4 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A3 @ I4 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10151_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10152_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat
        @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_10153_le__sequentially,axiom,
    ! [F6: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F6 @ at_top_nat )
      = ( ! [N11: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N11 ) @ F6 ) ) ) ).

% le_sequentially
thf(fact_10154_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N11: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N11 @ N )
           => ( P @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_10155_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C @ X3 )
         => ( P @ X3 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10156_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_10157_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10158_lhopital__left__at__top__at__top,axiom,
    ! [F: real > real,A3: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
thf(fact_10159_lhopital__at__top__at__top,axiom,
    ! [F: real > real,A3: real,G: real > real,F3: real > real,G2: real > real] :
      ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ at_top_real
                @ ( topolo2177554685111907308n_real @ A3 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top_at_top
thf(fact_10160_exp__at__top,axiom,
    filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).

% exp_at_top
thf(fact_10161_sqrt__at__top,axiom,
    filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).

% sqrt_at_top
thf(fact_10162_ln__at__top,axiom,
    filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).

% ln_at_top
thf(fact_10163_eventually__at__left__real,axiom,
    ! [B3: real,A3: real] :
      ( ( ord_less_real @ B3 @ A3 )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B3 @ A3 ) )
        @ ( topolo2177554685111907308n_real @ A3 @ ( set_or5984915006950818249n_real @ A3 ) ) ) ) ).

% eventually_at_left_real
thf(fact_10164_lhopital__left__at__top,axiom,
    ! [G: real > real,X2: real,G2: real > real,F: real > real,F3: real > real,Y2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ).

% lhopital_left_at_top
thf(fact_10165_lhospital__at__top__at__top,axiom,
    ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ at_top_real )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ at_top_real )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ at_top_real )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ at_top_real )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ X2 )
                @ at_top_real ) ) ) ) ) ) ).

% lhospital_at_top_at_top
thf(fact_10166_lhopital__at__top,axiom,
    ! [G: real > real,X2: real,G2: real > real,F: real > real,F3: real > real,Y2: real] :
      ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( eventually_real
          @ ^ [X: real] :
              ( ( G2 @ X )
             != zero_zero_real )
          @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
              @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
           => ( ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
             => ( filterlim_real_real
                @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                @ ( topolo2815343760600316023s_real @ Y2 )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).

% lhopital_at_top
thf(fact_10167_filterlim__real__sequentially,axiom,
    filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).

% filterlim_real_sequentially
thf(fact_10168_ln__x__over__x__tendsto__0,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ X ) @ X )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_real ) ).

% ln_x_over_x_tendsto_0
thf(fact_10169_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10170_lhopital,axiom,
    ! [F: real > real,X2: real,G: real > real,G2: real > real,F3: real > real,F6: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                    @ F6
                    @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F6
                    @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ) ).

% lhopital
thf(fact_10171_lhopital__left,axiom,
    ! [F: real > real,X2: real,G: real > real,G2: real > real,F3: real > real,F6: filter_real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
     => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
       => ( ( eventually_real
            @ ^ [X: real] :
                ( ( G @ X )
               != zero_zero_real )
            @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
         => ( ( eventually_real
              @ ^ [X: real] :
                  ( ( G2 @ X )
                 != zero_zero_real )
              @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
           => ( ( eventually_real
                @ ^ [X: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
             => ( ( eventually_real
                  @ ^ [X: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
                  @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
               => ( ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F3 @ X ) @ ( G2 @ X ) )
                    @ F6
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) )
                 => ( filterlim_real_real
                    @ ^ [X: real] : ( divide_divide_real @ ( F @ X ) @ ( G @ X ) )
                    @ F6
                    @ ( topolo2177554685111907308n_real @ X2 @ ( set_or5984915006950818249n_real @ X2 ) ) ) ) ) ) ) ) ) ) ).

% lhopital_left
thf(fact_10172_tendsto__exp__limit__at__top,axiom,
    ! [X2: real] :
      ( filterlim_real_real
      @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ Y ) ) @ Y )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X2 ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10173_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_10174_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_10175_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,F: real > real,Flim: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ B3 @ X3 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top

% Helper facts (58)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X2: int,Y2: int] :
      ( ( if_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( if_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y2: num] :
      ( ( if_num @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X2: num,Y2: num] :
      ( ( if_num @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( if_rat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X2: rat,Y2: rat] :
      ( ( if_rat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y2: real] :
      ( ( if_real @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X2: real,Y2: real] :
      ( ( if_real @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( if_uint32 @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Uint32__Ouint32_T,axiom,
    ! [X2: uint32,Y2: uint32] :
      ( ( if_uint32 @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X6: real] : ( P @ X6 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( if_complex @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X2: complex,Y2: complex] :
      ( ( if_complex @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y2: extended_enat] :
      ( ( if_Extended_enat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X2: extended_enat,Y2: extended_enat] :
      ( ( if_Extended_enat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( if_Code_integer @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X2: code_integer,Y2: code_integer] :
      ( ( if_Code_integer @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( if_set_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X2: set_int,Y2: set_int] :
      ( ( if_set_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X2: set_nat,Y2: set_nat] :
      ( ( if_set_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X2: vEBT_VEBT,Y2: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X2: list_int,Y2: list_int] :
      ( ( if_list_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y2: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( if_VEBT_VEBTi @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__VEBT____BuildupMemImp__OVEBTi_T,axiom,
    ! [X2: vEBT_VEBTi,Y2: vEBT_VEBTi] :
      ( ( if_VEBT_VEBTi @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y2: option_nat] :
      ( ( if_option_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X2: option_nat,Y2: option_nat] :
      ( ( if_option_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y2: option_num] :
      ( ( if_option_num @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X2: option_num,Y2: option_num] :
      ( ( if_option_num @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X2: heap_Time_Heap_o,Y2: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_I_Eo_J_T,axiom,
    ! [X2: heap_Time_Heap_o,Y2: heap_Time_Heap_o] :
      ( ( if_Heap_Time_Heap_o @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X2: heap_Time_Heap_nat,Y2: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Nat__Onat_J_T,axiom,
    ! [X2: heap_Time_Heap_nat,Y2: heap_Time_Heap_nat] :
      ( ( if_Hea2662716070787841314ap_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y2: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X2: product_prod_int_int,Y2: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X2: produc6271795597528267376eger_o,Y2: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__VEBT____BuildupMemImp__OVEBTi_J_T,axiom,
    ! [X2: heap_T8145700208782473153_VEBTi,Y2: heap_T8145700208782473153_VEBTi] :
      ( ( if_Hea8453224502484754311_VEBTi @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X2: produc827990862158126777uint32,Y2: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Uint32__Ouint32_Mt__Uint32__Ouint32_J_T,axiom,
    ! [X2: produc827990862158126777uint32,Y2: produc827990862158126777uint32] :
      ( ( if_Pro1135515155860407935uint32 @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X2: heap_T2636463487746394924on_nat,Y2: heap_T2636463487746394924on_nat] :
      ( ( if_Hea5867803462524415986on_nat @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__Option__Ooption_It__Nat__Onat_J_J_T,axiom,
    ! [X2: heap_T2636463487746394924on_nat,Y2: heap_T2636463487746394924on_nat] :
      ( ( if_Hea5867803462524415986on_nat @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Heap____Time____Monad__OHeap_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J_T,axiom,
    ! [X2: heap_T4980287057938770641_VEBTi,Y2: heap_T4980287057938770641_VEBTi] :
      ( ( if_Hea811341299636385687_VEBTi @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Heap____Time____Monad__OHeap_It__List__Olist_It__VEBT____BuildupMemImp__OVEBTi_J_J_T,axiom,
    ! [X2: heap_T4980287057938770641_VEBTi,Y2: heap_T4980287057938770641_VEBTi] :
      ( ( if_Hea811341299636385687_VEBTi @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X2: produc8923325533196201883nteger,Y2: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y2 )
      = X2 ) ).

thf(help_If_3_1_If_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_T,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( if_wor5778924947035936048l_num1 @ $false @ X2 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Word__Oword_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Obit0_It__Numeral____Type__Onum1_J_J_J_J_J_J_T,axiom,
    ! [X2: word_N3645301735248828278l_num1,Y2: word_N3645301735248828278l_num1] :
      ( ( if_wor5778924947035936048l_num1 @ $true @ X2 @ Y2 )
      = X2 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [T8: vEBT_VEBT,Ti5: vEBT_VEBTi,X4: nat] : ( refine7594492741263601813on_nat @ ( vEBT_vebt_predi @ Ti5 @ X4 ) @ ( f @ T8 @ Ti5 @ X4 ) ) ).

thf(conj_1,conjecture,
    ( refine7594492741263601813on_nat
    @ ( vEBT_c6250501799366334488on_nat
      @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
          ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
          @ ^ [Mima: product_prod_nat_nat] :
              ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
              @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  @ ^ [L: nat] :
                      ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                      @ ^ [H: nat] :
                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                          @ ^ [Aktnode: vEBT_VEBTi] :
                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                              @ ^ [Minlow: option_nat] :
                                  ( if_Hea5867803462524415986on_nat
                                  @ ( ( Minlow != none_nat )
                                    & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                  @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Aktnode @ L )
                                    @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                  @ ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_predi @ Summary2 @ H )
                                    @ ^ [Predsum: option_nat] :
                                        ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                        @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                          @ ^ [Nextnode: vEBT_VEBTi] :
                                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                              @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) )
          @ Info2 )
      @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ xa ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( xa = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
      @ tia )
    @ ( vEBT_c6250501799366334488on_nat
      @ ^ [Info2: option4927543243414619207at_nat,Deg2: nat,TreeArray2: array_VEBT_VEBTi,Summary2: vEBT_VEBTi] :
          ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( vEBT_is_Node @ ta ) )
          @ ^ [Uu: product_unit] :
              ( produc2190226783763740553on_nat
              @ ^ [Info3: option4927543243414619207at_nat] :
                  ( produc2606485630176857543on_nat
                  @ ^ [Deg3: nat] :
                      ( produc6124225815318652422on_nat
                      @ ^ [TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( heap_T5999496708990702694on_nat
                          @ ( refine_Imp_assert
                            @ ( ( Info3 = Info2 )
                              & ( Deg3 = Deg2 )
                              & ( vEBT_is_Node @ ta ) ) )
                          @ ^ [Uv: product_unit] :
                              ( case_o8344607093967974880at_nat @ ( heap_T3487192422709364219on_nat @ none_nat )
                              @ ^ [Mima: product_prod_nat_nat] :
                                  ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ Deg2 @ one_one_nat ) @ ( heap_T3487192422709364219on_nat @ none_nat )
                                  @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_snd_nat_nat @ Mima ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_snd_nat_nat @ Mima ) ) )
                                    @ ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_lowi @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                      @ ^ [L: nat] :
                                          ( heap_T8222160169144143993on_nat @ ( vEBT_VEBT_highi @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                                          @ ^ [H: nat] :
                                              ( heap_T5999496708990702694on_nat
                                              @ ( refine_Imp_assert
                                                @ ( L
                                                  = ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                              @ ^ [Uw: product_unit] :
                                                  ( heap_T5999496708990702694on_nat
                                                  @ ( refine_Imp_assert
                                                    @ ( H
                                                      = ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                  @ ^ [Ux: product_unit] :
                                                      ( heap_T5999496708990702694on_nat @ ( refine_Imp_assert @ ( ord_less_nat @ H @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) )
                                                      @ ^ [Uy: product_unit] :
                                                          ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ H )
                                                          @ ^ [Aktnode: vEBT_VEBTi] :
                                                              ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_minti @ Aktnode )
                                                              @ ^ [Minlow: option_nat] :
                                                                  ( heap_T5999496708990702694on_nat
                                                                  @ ( refine_Imp_assert
                                                                    @ ( Minlow
                                                                      = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ H ) ) ) )
                                                                  @ ^ [Uz: product_unit] :
                                                                      ( if_Hea5867803462524415986on_nat
                                                                      @ ( ( Minlow != none_nat )
                                                                        & ( vEBT_VEBT_greater @ ( some_nat @ L ) @ Minlow ) )
                                                                      @ ( heap_T3669509953089699273on_nat @ ( f @ ( nth_VEBT_VEBT @ TreeList @ H ) @ Aktnode @ L )
                                                                        @ ^ [Predy: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ H ) ) @ Predy ) ) )
                                                                      @ ( heap_T3669509953089699273on_nat @ ( f @ Summary3 @ Summary2 @ H )
                                                                        @ ^ [Predsum: option_nat] :
                                                                            ( heap_T5999496708990702694on_nat
                                                                            @ ( refine_Imp_assert
                                                                              @ ( ( Predsum = none_nat )
                                                                                = ( ( vEBT_vebt_pred @ Summary3 @ H )
                                                                                  = none_nat ) ) )
                                                                            @ ^ [Va: product_unit] :
                                                                                ( if_Hea5867803462524415986on_nat @ ( Predsum = none_nat ) @ ( if_Hea5867803462524415986on_nat @ ( ord_less_nat @ ( product_fst_nat_nat @ Mima ) @ xa ) @ ( heap_T3487192422709364219on_nat @ ( some_nat @ ( product_fst_nat_nat @ Mima ) ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) )
                                                                                @ ( heap_T2868974464944644318on_nat @ ( array_nth_VEBT_VEBTi @ TreeArray2 @ ( the_nat @ Predsum ) )
                                                                                  @ ^ [Nextnode: vEBT_VEBTi] :
                                                                                      ( heap_T3669509953089699273on_nat @ ( vEBT_vebt_maxti @ Nextnode )
                                                                                      @ ^ [Maxnext: option_nat] : ( heap_T3487192422709364219on_nat @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Predsum ) @ Maxnext ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
                              @ Info2 ) ) ) )
              @ ( vEBT_c634343235235684882T_VEBT
                @ ^ [Info3: option4927543243414619207at_nat,Deg3: nat,TreeList: list_VEBT_VEBT,Summary3: vEBT_VEBT] : ( produc6272728841373537334T_VEBT @ Info3 @ ( produc1750349459881913976T_VEBT @ Deg3 @ ( produc6691630295680060561T_VEBT @ TreeList @ Summary3 ) ) )
                @ ^ [A2: $o,B2: $o] : undefi7074909574607331924T_VEBT
                @ ta ) ) )
      @ ^ [A2: $o,B2: $o] : ( if_Hea5867803462524415986on_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ xa ) @ ( if_Hea5867803462524415986on_nat @ B2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ one_one_nat ) ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) ) @ ( if_Hea5867803462524415986on_nat @ ( xa = one_one_nat ) @ ( if_Hea5867803462524415986on_nat @ A2 @ ( heap_T3487192422709364219on_nat @ ( some_nat @ zero_zero_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) @ ( heap_T3487192422709364219on_nat @ none_nat ) ) )
      @ tia ) ) ).

%------------------------------------------------------------------------------